Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 98.7%

Time: 15.0s

Alternatives: 24

Speedup: 1.8×

• 52.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := 0.375 + v \cdot -0.25\\ \mathbf{if}\;r\_m \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + \left(-1.5 + t\_0 \cdot \frac{w \cdot \left(\left(r\_m \cdot r\_m\right) \cdot w\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_0 \cdot \frac{\left(r\_m \cdot w\right) \cdot \left(r\_m \cdot w\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (+ 0.375 (* v -0.25))))
   (if (<= r_m 1.4e-10)
     (+
      (/ 2.0 (* r_m r_m))
      (+ -1.5 (* t_0 (/ (* w (* (* r_m r_m) w)) (+ v -1.0)))))
     (- (+ 3.0 (* t_0 (/ (* (* r_m w) (* r_m w)) (+ v -1.0)))) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 0.375 + (v * -0.25);
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (t_0 * ((w * ((r_m * r_m) * w)) / (v + -1.0))));
	} else {
		tmp = (3.0 + (t_0 * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.375d0 + (v * (-0.25d0))
    if (r_m <= 1.4d-10) then
        tmp = (2.0d0 / (r_m * r_m)) + ((-1.5d0) + (t_0 * ((w * ((r_m * r_m) * w)) / (v + (-1.0d0)))))
    else
        tmp = (3.0d0 + (t_0 * (((r_m * w) * (r_m * w)) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 0.375 + (v * -0.25);
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (t_0 * ((w * ((r_m * r_m) * w)) / (v + -1.0))));
	} else {
		tmp = (3.0 + (t_0 * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 0.375 + (v * -0.25)
	tmp = 0
	if r_m <= 1.4e-10:
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (t_0 * ((w * ((r_m * r_m) * w)) / (v + -1.0))))
	else:
		tmp = (3.0 + (t_0 * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(0.375 + Float64(v * -0.25))
	tmp = 0.0
	if (r_m <= 1.4e-10)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-1.5 + Float64(t_0 * Float64(Float64(w * Float64(Float64(r_m * r_m) * w)) / Float64(v + -1.0)))));
	else
		tmp = Float64(Float64(3.0 + Float64(t_0 * Float64(Float64(Float64(r_m * w) * Float64(r_m * w)) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 0.375 + (v * -0.25);
	tmp = 0.0;
	if (r_m <= 1.4e-10)
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (t_0 * ((w * ((r_m * r_m) * w)) / (v + -1.0))));
	else
		tmp = (3.0 + (t_0 * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 1.4e-10], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(t$95$0 * N[(N[(w * N[(N[(r$95$m * r$95$m), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(t$95$0 * N[(N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.40000000000000008e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. *-lowering-*.f64 95.6%

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

   6. Applied egg-rr 95.6%

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

   if 1.40000000000000008e-10 < r

   1. Initial program 91.3%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

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

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

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

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

   7. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \color{blue}{\frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v} \cdot \left(0.375 + 0.125 \cdot \left(v \cdot -2\right)\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

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

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

      4. *-lowering-*.f64 99.8%

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

   9. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{1 - v} \cdot \left(0.375 + 0.125 \cdot \left(v \cdot -2\right)\right)\right) - 4.5 \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

       4. metadata-eval N/A

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

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

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

       6. metadata-eval 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

   11. Applied egg-rr 99.8%

       \[\leadsto \left(3 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v} \cdot \left(0.375 + \color{blue}{v \cdot -0.25}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(\left(r \cdot r\right) \cdot w\right)}{v + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m} + \left(-1.5 + \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{elif}\;r\_m \leq 1.38 \cdot 10^{+165}:\\ \;\;\;\;\left(3 + \left(r\_m \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r\_m \cdot \left(w \cdot w\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.4e-10)
   (+ (/ (/ 2.0 r_m) r_m) (+ -1.5 (* (* r_m w) (* (* r_m w) -0.375))))
   (if (<= r_m 1.38e+165)
     (-
      (+ 3.0 (* (* r_m (+ 0.375 (* v -0.25))) (/ (* r_m (* w w)) (+ v -1.0))))
      4.5)
     (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w)))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	} else if (r_m <= 1.38e+165) {
		tmp = (3.0 + ((r_m * (0.375 + (v * -0.25))) * ((r_m * (w * w)) / (v + -1.0)))) - 4.5;
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.4d-10) then
        tmp = ((2.0d0 / r_m) / r_m) + ((-1.5d0) + ((r_m * w) * ((r_m * w) * (-0.375d0))))
    else if (r_m <= 1.38d+165) then
        tmp = (3.0d0 + ((r_m * (0.375d0 + (v * (-0.25d0)))) * ((r_m * (w * w)) / (v + (-1.0d0))))) - 4.5d0
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	} else if (r_m <= 1.38e+165) {
		tmp = (3.0 + ((r_m * (0.375 + (v * -0.25))) * ((r_m * (w * w)) / (v + -1.0)))) - 4.5;
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.4e-10:
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)))
	elif r_m <= 1.38e+165:
		tmp = (3.0 + ((r_m * (0.375 + (v * -0.25))) * ((r_m * (w * w)) / (v + -1.0)))) - 4.5
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.4e-10)
		tmp = Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 + Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * -0.375))));
	elseif (r_m <= 1.38e+165)
		tmp = Float64(Float64(3.0 + Float64(Float64(r_m * Float64(0.375 + Float64(v * -0.25))) * Float64(Float64(r_m * Float64(w * w)) / Float64(v + -1.0)))) - 4.5);
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.4e-10)
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	elseif (r_m <= 1.38e+165)
		tmp = (3.0 + ((r_m * (0.375 + (v * -0.25))) * ((r_m * (w * w)) / (v + -1.0)))) - 4.5;
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.4e-10], N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 + N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r$95$m, 1.38e+165], N[(N[(3.0 + N[(N[(r$95$m * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r$95$m * N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.40000000000000008e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.5%

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

   7. Simplified 81.5%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 92.0%

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

   9. Applied egg-rr 92.0%

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

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

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

       2. associate-/r* N/A

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

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

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

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

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

       5. +-commutative N/A

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

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

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 94.1%

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

   11. Applied egg-rr 94.1%

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

   if 1.40000000000000008e-10 < r < 1.37999999999999997e165

   1. Initial program 99.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 99.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

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

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

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

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

   7. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \color{blue}{\frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v} \cdot \left(0.375 + 0.125 \cdot \left(v \cdot -2\right)\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\left(\left(\frac{3}{8} + v \cdot \frac{-1}{4}\right) \cdot r\right), \left(\frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\left(r \cdot \left(\frac{3}{8} + v \cdot \frac{-1}{4}\right)\right), \left(\frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right), \left(\frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \left(\frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right)\right), \frac{9}{2}\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right), \mathsf{/.f64}\left(\left(w \cdot \left(r \cdot w\right)\right), \left(1 - v\right)\right)\right)\right), \frac{9}{2}\right) \]

   9. Applied egg-rr 94.2%

      \[\leadsto \left(3 - \color{blue}{\left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}}\right) - 4.5 \]

   if 1.37999999999999997e165 < r

   1. Initial program 82.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 82.8%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.3%

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

   8. Simplified 70.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 93.2%

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

   10. Applied egg-rr 93.2%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{elif}\;r \leq 1.38 \cdot 10^{+165}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{elif}\;r\_m \leq 1.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + \left(-1.5 + \left(r\_m \cdot r\_m\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.15e-100)
   (/ (/ 2.0 r_m) r_m)
   (if (<= r_m 1.6e+100)
     (+ (/ 2.0 (* r_m r_m)) (+ -1.5 (* (* r_m r_m) (* -0.375 (* w w)))))
     (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w)))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.15e-100) {
		tmp = (2.0 / r_m) / r_m;
	} else if (r_m <= 1.6e+100) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * r_m) * (-0.375 * (w * w))));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.15d-100) then
        tmp = (2.0d0 / r_m) / r_m
    else if (r_m <= 1.6d+100) then
        tmp = (2.0d0 / (r_m * r_m)) + ((-1.5d0) + ((r_m * r_m) * ((-0.375d0) * (w * w))))
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.15e-100) {
		tmp = (2.0 / r_m) / r_m;
	} else if (r_m <= 1.6e+100) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * r_m) * (-0.375 * (w * w))));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.15e-100:
		tmp = (2.0 / r_m) / r_m
	elif r_m <= 1.6e+100:
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * r_m) * (-0.375 * (w * w))))
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.15e-100)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	elseif (r_m <= 1.6e+100)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-1.5 + Float64(Float64(r_m * r_m) * Float64(-0.375 * Float64(w * w)))));
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.15e-100)
		tmp = (2.0 / r_m) / r_m;
	elseif (r_m <= 1.6e+100)
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * r_m) * (-0.375 * (w * w))));
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.15e-100], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], If[LessEqual[r$95$m, 1.6e+100], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.14999999999999997e-100

   1. Initial program 85.6%

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

      1. associate--l- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   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 57.6%

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

   7. Simplified 57.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 57.7%

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

   9. Applied egg-rr 57.7%

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

   if 1.14999999999999997e-100 < r < 1.5999999999999999e100

   1. Initial program 96.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 99.6%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 90.9%

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

   7. Simplified 90.9%

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

   if 1.5999999999999999e100 < r

   1. Initial program 87.6%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 87.6%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 74.8%

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

   8. Simplified 74.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 91.9%

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

   10. Applied egg-rr 91.9%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m} + \left(-1.5 + \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{\left(r\_m \cdot w\right) \cdot \left(r\_m \cdot w\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 9.2e-11)
   (+ (/ (/ 2.0 r_m) r_m) (+ -1.5 (* (* r_m w) (* (* r_m w) -0.375))))
   (-
    (+ 3.0 (* (+ 0.375 (* v -0.25)) (/ (* (* r_m w) (* r_m w)) (+ v -1.0))))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 9.2e-11) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	} else {
		tmp = (3.0 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 9.2d-11) then
        tmp = ((2.0d0 / r_m) / r_m) + ((-1.5d0) + ((r_m * w) * ((r_m * w) * (-0.375d0))))
    else
        tmp = (3.0d0 + ((0.375d0 + (v * (-0.25d0))) * (((r_m * w) * (r_m * w)) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 9.2e-11) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	} else {
		tmp = (3.0 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 9.2e-11:
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)))
	else:
		tmp = (3.0 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 9.2e-11)
		tmp = Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 + Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * -0.375))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(Float64(r_m * w) * Float64(r_m * w)) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 9.2e-11)
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	else
		tmp = (3.0 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 9.2e-11], N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 + N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.20000000000000054e-11

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.5%

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

   7. Simplified 81.5%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 92.0%

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

   9. Applied egg-rr 92.0%

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

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

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

       2. associate-/r* N/A

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

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

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

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

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

       5. +-commutative N/A

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

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

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 94.1%

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

   11. Applied egg-rr 94.1%

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

   if 9.20000000000000054e-11 < r

   1. Initial program 91.3%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

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

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

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

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

   7. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \color{blue}{\frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v} \cdot \left(0.375 + 0.125 \cdot \left(v \cdot -2\right)\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

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

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

      4. *-lowering-*.f64 99.8%

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

   9. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}{1 - v} \cdot \left(0.375 + 0.125 \cdot \left(v \cdot -2\right)\right)\right) - 4.5 \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

       4. metadata-eval N/A

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

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

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

       6. metadata-eval 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

   11. Applied egg-rr 99.8%

       \[\leadsto \left(3 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v} \cdot \left(0.375 + \color{blue}{v \cdot -0.25}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m} + \left(-1.5 + \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r\_m \cdot \left(w \cdot \left(r\_m \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.4e-10)
   (+ (/ (/ 2.0 r_m) r_m) (+ -1.5 (* (* r_m w) (* (* r_m w) -0.375))))
   (-
    (+ 3.0 (* (+ 0.375 (* v -0.25)) (/ (* r_m (* w (* r_m w))) (+ v -1.0))))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	} else {
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.4d-10) then
        tmp = ((2.0d0 / r_m) / r_m) + ((-1.5d0) + ((r_m * w) * ((r_m * w) * (-0.375d0))))
    else
        tmp = (3.0d0 + ((0.375d0 + (v * (-0.25d0))) * ((r_m * (w * (r_m * w))) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	} else {
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.4e-10:
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)))
	else:
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.4e-10)
		tmp = Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 + Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * -0.375))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r_m * Float64(w * Float64(r_m * w))) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.4e-10)
		tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
	else
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.4e-10], N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 + N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r$95$m * N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.40000000000000008e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 81.5%

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

   7. Simplified 81.5%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 92.0%

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

   9. Applied egg-rr 92.0%

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

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

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

       2. associate-/r* N/A

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

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

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

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

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

       5. +-commutative N/A

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

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

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 94.1%

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

   11. Applied egg-rr 94.1%

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

   if 1.40000000000000008e-10 < r

   1. Initial program 91.3%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. swap-sqr N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

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

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

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

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

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

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

      17. sub-neg N/A

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

      18. distribute-lft-in N/A

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

      19. metadata-eval N/A

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

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

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

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

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

   7. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \color{blue}{\frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v} \cdot \left(0.375 + 0.125 \cdot \left(v \cdot -2\right)\right)}\right) - 4.5 \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \left(v \cdot \frac{-1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

      4. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{+.f64}\left(\frac{3}{8}, \mathsf{*.f64}\left(v, \frac{-1}{4}\right)\right)\right)\right), \frac{9}{2}\right) \]

   9. Applied egg-rr 99.8%

      \[\leadsto \left(3 - \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v} \cdot \left(0.375 + \color{blue}{v \cdot -0.25}\right)\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{if}\;r\_m \leq 5 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;r\_m \leq 5600000:\\ \;\;\;\;t\_0 + \left(r\_m \cdot r\_m\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r_m) r_m)))
   (if (<= r_m 5e-101)
     t_0
     (if (<= r_m 5600000.0)
       (+ t_0 (* (* r_m r_m) (* -0.375 (* w w))))
       (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (2.0 / r_m) / r_m;
	double tmp;
	if (r_m <= 5e-101) {
		tmp = t_0;
	} else if (r_m <= 5600000.0) {
		tmp = t_0 + ((r_m * r_m) * (-0.375 * (w * w)));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / r_m) / r_m
    if (r_m <= 5d-101) then
        tmp = t_0
    else if (r_m <= 5600000.0d0) then
        tmp = t_0 + ((r_m * r_m) * ((-0.375d0) * (w * w)))
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = (2.0 / r_m) / r_m;
	double tmp;
	if (r_m <= 5e-101) {
		tmp = t_0;
	} else if (r_m <= 5600000.0) {
		tmp = t_0 + ((r_m * r_m) * (-0.375 * (w * w)));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = (2.0 / r_m) / r_m
	tmp = 0
	if r_m <= 5e-101:
		tmp = t_0
	elif r_m <= 5600000.0:
		tmp = t_0 + ((r_m * r_m) * (-0.375 * (w * w)))
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(2.0 / r_m) / r_m)
	tmp = 0.0
	if (r_m <= 5e-101)
		tmp = t_0;
	elseif (r_m <= 5600000.0)
		tmp = Float64(t_0 + Float64(Float64(r_m * r_m) * Float64(-0.375 * Float64(w * w))));
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = (2.0 / r_m) / r_m;
	tmp = 0.0;
	if (r_m <= 5e-101)
		tmp = t_0;
	elseif (r_m <= 5600000.0)
		tmp = t_0 + ((r_m * r_m) * (-0.375 * (w * w)));
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]}, If[LessEqual[r$95$m, 5e-101], t$95$0, If[LessEqual[r$95$m, 5600000.0], N[(t$95$0 + N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.0000000000000001e-101

   1. Initial program 85.6%

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

      1. associate--l- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   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 57.6%

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

   7. Simplified 57.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 57.7%

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

   9. Applied egg-rr 57.7%

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

   if 5.0000000000000001e-101 < r < 5.6e6

   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 99.5%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 92.9%

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

   7. Simplified 92.9%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

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

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-lowering-*.f64 92.9%

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

   9. Applied egg-rr 92.9%

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

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

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

       2. associate-/r* N/A

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

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

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

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

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

       5. +-commutative N/A

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

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

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

       7. associate-*r* N/A

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

       8. associate-*l* N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 92.6%

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

   11. Applied egg-rr 92.6%

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

   12. Taylor expanded in r around inf

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 92.6%

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

   14. Simplified 92.6%

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

   if 5.6e6 < r

   1. Initial program 91.3%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 87.4%

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

   10. Applied egg-rr 87.4%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 5600000:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 7.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{elif}\;r\_m \leq 5600000:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + \left(r\_m \cdot r\_m\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 7.8e-101)
   (/ (/ 2.0 r_m) r_m)
   (if (<= r_m 5600000.0)
     (+ (/ 2.0 (* r_m r_m)) (* (* r_m r_m) (* -0.375 (* w w))))
     (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w)))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.8e-101) {
		tmp = (2.0 / r_m) / r_m;
	} else if (r_m <= 5600000.0) {
		tmp = (2.0 / (r_m * r_m)) + ((r_m * r_m) * (-0.375 * (w * w)));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 7.8d-101) then
        tmp = (2.0d0 / r_m) / r_m
    else if (r_m <= 5600000.0d0) then
        tmp = (2.0d0 / (r_m * r_m)) + ((r_m * r_m) * ((-0.375d0) * (w * w)))
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.8e-101) {
		tmp = (2.0 / r_m) / r_m;
	} else if (r_m <= 5600000.0) {
		tmp = (2.0 / (r_m * r_m)) + ((r_m * r_m) * (-0.375 * (w * w)));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 7.8e-101:
		tmp = (2.0 / r_m) / r_m
	elif r_m <= 5600000.0:
		tmp = (2.0 / (r_m * r_m)) + ((r_m * r_m) * (-0.375 * (w * w)))
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 7.8e-101)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	elseif (r_m <= 5600000.0)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(Float64(r_m * r_m) * Float64(-0.375 * Float64(w * w))));
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 7.8e-101)
		tmp = (2.0 / r_m) / r_m;
	elseif (r_m <= 5600000.0)
		tmp = (2.0 / (r_m * r_m)) + ((r_m * r_m) * (-0.375 * (w * w)));
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 7.8e-101], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], If[LessEqual[r$95$m, 5600000.0], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.80000000000000031e-101

   1. Initial program 85.6%

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

      1. associate--l- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   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 57.6%

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

   7. Simplified 57.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 57.7%

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

   9. Applied egg-rr 57.7%

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

   if 7.80000000000000031e-101 < r < 5.6e6

   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 99.5%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 92.9%

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

   10. Simplified 92.9%

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

   if 5.6e6 < r

   1. Initial program 91.3%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 75.6%

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

   8. Simplified 75.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 87.4%

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

   10. Applied egg-rr 87.4%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 5600000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{\left(r\_m \cdot w\right) \cdot \left(r\_m \cdot w\right)}{v + -1}\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+
  (/ 2.0 (* r_m r_m))
  (+ -1.5 (* (+ 0.375 (* v -0.25)) (/ (* (* r_m w) (* r_m w)) (+ v -1.0))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0))));
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = (2.0d0 / (r_m * r_m)) + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * (((r_m * w) * (r_m * w)) / (v + (-1.0d0)))))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0))));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0))))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(Float64(r_m * w) * Float64(r_m * w)) / Float64(v + -1.0)))))
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * (((r_m * w) * (r_m * w)) / (v + -1.0))));
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

   1. associate--l- N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

   7. associate--r+ N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

3. Simplified 97.9%

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

   1. associate-*r* N/A

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

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

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

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

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

   4. *-lowering-*.f64 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 9: 97.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r\_m \cdot \left(w \cdot \left(r\_m \cdot w\right)\right)}{v + -1}\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+
  (/ 2.0 (* r_m r_m))
  (+ -1.5 (* (+ 0.375 (* v -0.25)) (/ (* r_m (* w (* r_m w))) (+ v -1.0))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0))));
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = (2.0d0 / (r_m * r_m)) + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * ((r_m * (w * (r_m * w))) / (v + (-1.0d0)))))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0))));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0))))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r_m * Float64(w * Float64(r_m * w))) / Float64(v + -1.0)))))
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((0.375 + (v * -0.25)) * ((r_m * (w * (r_m * w))) / (v + -1.0))));
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r$95$m * N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.5%

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

   1. associate--l- N/A

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

   2. +-commutative N/A

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

   3. associate--l+ N/A

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

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

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

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

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

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

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

   7. associate--r+ N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

3. Simplified 97.9%

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

5. Final simplification 97.9%

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

Alternative 10: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 5.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + \left(-1.5 + w \cdot \left(-0.375 \cdot \left(r\_m \cdot \left(r\_m \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 5.7e+163)
   (+ (/ 2.0 (* r_m r_m)) (+ -1.5 (* w (* -0.375 (* r_m (* r_m w))))))
   (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 5.7e+163) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (w * (-0.375 * (r_m * (r_m * w)))));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 5.7d+163) then
        tmp = (2.0d0 / (r_m * r_m)) + ((-1.5d0) + (w * ((-0.375d0) * (r_m * (r_m * w)))))
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 5.7e+163) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (w * (-0.375 * (r_m * (r_m * w)))));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 5.7e+163:
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (w * (-0.375 * (r_m * (r_m * w)))))
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 5.7e+163)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-1.5 + Float64(w * Float64(-0.375 * Float64(r_m * Float64(r_m * w))))));
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 5.7e+163)
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + (w * (-0.375 * (r_m * (r_m * w)))));
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 5.7e+163], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(w * N[(-0.375 * N[(r$95$m * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.6999999999999998e163

   1. Initial program 88.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.6%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 82.6%

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

   7. Simplified 82.6%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

       6. *-lowering-*.f64 93.6%

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

   11. Applied egg-rr 93.6%

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

   if 5.6999999999999998e163 < r

   1. Initial program 82.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 82.8%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 70.3%

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

   8. Simplified 70.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 93.2%

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

   10. Applied egg-rr 93.2%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + w \cdot \left(-0.375 \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 7.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + \left(-1.5 + \left(r\_m \cdot \left(r\_m \cdot w\right)\right) \cdot \left(w \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 7.2e+166)
   (+ (/ 2.0 (* r_m r_m)) (+ -1.5 (* (* r_m (* r_m w)) (* w -0.375))))
   (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.2e+166) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * (r_m * w)) * (w * -0.375)));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 7.2d+166) then
        tmp = (2.0d0 / (r_m * r_m)) + ((-1.5d0) + ((r_m * (r_m * w)) * (w * (-0.375d0))))
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.2e+166) {
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * (r_m * w)) * (w * -0.375)));
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 7.2e+166:
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * (r_m * w)) * (w * -0.375)))
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 7.2e+166)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-1.5 + Float64(Float64(r_m * Float64(r_m * w)) * Float64(w * -0.375))));
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 7.2e+166)
		tmp = (2.0 / (r_m * r_m)) + (-1.5 + ((r_m * (r_m * w)) * (w * -0.375)));
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 7.2e+166], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r$95$m * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.1999999999999994e166

   1. Initial program 88.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.6%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 82.6%

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

   7. Simplified 82.6%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*r* N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 93.6%

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

   9. Applied egg-rr 93.6%

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

   if 7.1999999999999994e166 < r

   1. Initial program 82.8%

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

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 82.8%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-in N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 70.3%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   8. Simplified 70.3%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot \left(w \cdot \left(w \cdot \frac{-1}{4}\right)\right)\right), r\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\frac{-1}{4} \cdot w\right)\right), r\right)\right) \]

      9. *-lowering-*.f64 93.2%

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\frac{-1}{4}, w\right)\right), r\right)\right) \]

   10. Applied egg-rr 93.2%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.2 \cdot 10^{+166}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot \left(r \cdot w\right)\right) \cdot \left(w \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{\frac{2}{r\_m}}{r\_m} + \left(-1.5 + \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot -0.375\right)\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (+ (/ (/ 2.0 r_m) r_m) (+ -1.5 (* (* r_m w) (* (* r_m w) -0.375)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = ((2.0d0 / r_m) / r_m) + ((-1.5d0) + ((r_m * w) * ((r_m * w) * (-0.375d0))))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(2.0 / r_m) / r_m) + Float64(-1.5 + Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * -0.375))))
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = ((2.0 / r_m) / r_m) + (-1.5 + ((r_m * w) * ((r_m * w) * -0.375)));
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision] + N[(-1.5 + N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.5%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Step-by-step derivation

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   8. *-lowering-*.f64 80.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

7. Simplified 80.9%

   \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   2. swap-sqr N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(w \cdot r\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   11. *-lowering-*.f64 91.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

9. Applied egg-rr 91.3%

   \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375} + -1.5\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-3}{8} + \frac{-3}{2}\right)}\right) \]

    2. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\left(\frac{\frac{2}{r}}{r}\right), \left(\color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-3}{8}} + \frac{-3}{2}\right)\right) \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{r}\right), r\right), \left(\color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-3}{8}} + \frac{-3}{2}\right)\right) \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \left(\color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)} \cdot \frac{-3}{8} + \frac{-3}{2}\right)\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \left(\frac{-3}{2} + \color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-3}{8}}\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{-3}{8}\right)}\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right)\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)}\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot w\right), \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\color{blue}{\left(r \cdot w\right)} \cdot \frac{-3}{8}\right)\right)\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\left(r \cdot w\right), \color{blue}{\frac{-3}{8}}\right)\right)\right)\right) \]

    12. *-lowering-*.f64 92.8%

        \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \frac{-3}{8}\right)\right)\right)\right) \]

11. Applied egg-rr 92.8%

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)} \]
12. Add Preprocessing

Alternative 13: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r\_m \cdot \left(\left(r\_m \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.3e-11)
   (/ (/ 2.0 r_m) r_m)
   (+ -1.5 (* r_m (* (* r_m w) (* -0.25 w))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.3e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.3d-11) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (-1.5d0) + (r_m * ((r_m * w) * ((-0.25d0) * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.3e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.3e-11:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.3e-11)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(-1.5 + Float64(r_m * Float64(Float64(r_m * w) * Float64(-0.25 * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.3e-11)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = -1.5 + (r_m * ((r_m * w) * (-0.25 * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.3e-11], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(-1.5 + N[(r$95$m * N[(N[(r$95$m * w), $MachinePrecision] * N[(-0.25 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.3e-11

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.3e-11 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{-1}{4} \cdot \left(\color{blue}{{r}^{2}} \cdot {w}^{2}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 75.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   8. Simplified 75.6%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(r \cdot \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{-1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot \left(w \cdot \left(w \cdot \frac{-1}{4}\right)\right)\right), r\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \left(w \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(w \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(\frac{-1}{4} \cdot w\right)\right), r\right)\right) \]

      9. *-lowering-*.f64 87.4%

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{*.f64}\left(\frac{-1}{4}, w\right)\right), r\right)\right) \]

   10. Applied egg-rr 87.4%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(\left(r \cdot w\right) \cdot \left(-0.25 \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r\_m \cdot w\right) \cdot \left(\left(r\_m \cdot w\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.4e-10)
   (/ (/ 2.0 r_m) r_m)
   (+ -1.5 (* (* r_m w) (* (* r_m w) -0.375)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + ((r_m * w) * ((r_m * w) * -0.375));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.4d-10) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (-1.5d0) + ((r_m * w) * ((r_m * w) * (-0.375d0)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + ((r_m * w) * ((r_m * w) * -0.375));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.4e-10:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = -1.5 + ((r_m * w) * ((r_m * w) * -0.375))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.4e-10)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(-1.5 + Float64(Float64(r_m * w) * Float64(Float64(r_m * w) * -0.375)));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.4e-10)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = -1.5 + ((r_m * w) * ((r_m * w) * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.4e-10], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(-1.5 + N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.40000000000000008e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.40000000000000008e-10 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{-3}{8} \cdot \left(\color{blue}{{r}^{2}} \cdot {w}^{2}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \left(\frac{-3}{8} \cdot \color{blue}{{w}^{2}}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right)\right) \]

      16. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right)\right) \]

   8. Simplified 79.4%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

      2. swap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right), \color{blue}{\left(r \cdot w\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(r \cdot w\right), \frac{-3}{8}\right), \left(\color{blue}{r} \cdot w\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \frac{-3}{8}\right), \left(r \cdot w\right)\right)\right) \]

      8. *-lowering-*.f64 89.5%

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), \frac{-3}{8}\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right)\right) \]

   10. Applied egg-rr 89.5%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot w\right) \cdot -0.375\right) \cdot \left(r \cdot w\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 9.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + -0.375 \cdot \left(w \cdot \left(r\_m \cdot \left(r\_m \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 9.8e-11)
   (/ (/ 2.0 r_m) r_m)
   (+ -1.5 (* -0.375 (* w (* r_m (* r_m w)))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 9.8e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + (-0.375 * (w * (r_m * (r_m * w))));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 9.8d-11) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (-1.5d0) + ((-0.375d0) * (w * (r_m * (r_m * w))))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 9.8e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + (-0.375 * (w * (r_m * (r_m * w))));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 9.8e-11:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = -1.5 + (-0.375 * (w * (r_m * (r_m * w))))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 9.8e-11)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(-1.5 + Float64(-0.375 * Float64(w * Float64(r_m * Float64(r_m * w)))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 9.8e-11)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = -1.5 + (-0.375 * (w * (r_m * (r_m * w))));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 9.8e-11], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(-1.5 + N[(-0.375 * N[(w * N[(r$95$m * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.7999999999999998e-11

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 9.7999999999999998e-11 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{-3}{8} \cdot \left(\color{blue}{{r}^{2}} \cdot {w}^{2}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \left(\frac{-3}{8} \cdot \color{blue}{{w}^{2}}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right)\right) \]

      16. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right)\right) \]

   8. Simplified 79.4%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

      2. swap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right), \color{blue}{\frac{-3}{8}}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\left(w \cdot r\right) \cdot \left(r \cdot w\right)\right), \frac{-3}{8}\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \left(r \cdot \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right)\right) \]

      8. *-lowering-*.f64 87.8%

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{-3}{8}\right)\right) \]

   10. Applied egg-rr 87.8%

       \[\leadsto -1.5 + \color{blue}{\left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + -0.375 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r\_m \cdot r\_m\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.05e-10)
   (/ (/ 2.0 r_m) r_m)
   (+ -1.5 (* (* r_m r_m) (* -0.25 (* w w))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.05e-10) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + ((r_m * r_m) * (-0.25 * (w * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.05d-10) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (-1.5d0) + ((r_m * r_m) * ((-0.25d0) * (w * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.05e-10) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + ((r_m * r_m) * (-0.25 * (w * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.05e-10:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = -1.5 + ((r_m * r_m) * (-0.25 * (w * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.05e-10)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(-1.5 + Float64(Float64(r_m * r_m) * Float64(-0.25 * Float64(w * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.05e-10)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = -1.5 + ((r_m * r_m) * (-0.25 * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.05e-10], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(-1.5 + N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.05e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.05e-10 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{-1}{4} \cdot \left(\color{blue}{{r}^{2}} \cdot {w}^{2}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 75.6%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   8. Simplified 75.6%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 80.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r\_m \cdot r\_m\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 4.5e-11)
   (/ (/ 2.0 r_m) r_m)
   (+ -1.5 (* (* r_m r_m) (* -0.375 (* w w))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4.5e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + ((r_m * r_m) * (-0.375 * (w * w)));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 4.5d-11) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (-1.5d0) + ((r_m * r_m) * ((-0.375d0) * (w * w)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4.5e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5 + ((r_m * r_m) * (-0.375 * (w * w)));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 4.5e-11:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = -1.5 + ((r_m * r_m) * (-0.375 * (w * w)))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 4.5e-11)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(-1.5 + Float64(Float64(r_m * r_m) * Float64(-0.375 * Float64(w * w))));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 4.5e-11)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = -1.5 + ((r_m * r_m) * (-0.375 * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 4.5e-11], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(-1.5 + N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.5e-11

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 4.5e-11 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{-3}{8} \cdot \left(\color{blue}{{r}^{2}} \cdot {w}^{2}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \left(\frac{-3}{8} \cdot \color{blue}{{w}^{2}}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right)\right) \]

      16. *-lowering-*.f64 79.4%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right)\right) \]

   8. Simplified 79.4%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 8 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r\_m \cdot r\_m\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 8e+18)
   (+ (/ 2.0 (* r_m r_m)) -1.5)
   (* (* r_m r_m) (* -0.25 (* w w)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 8e+18) {
		tmp = (2.0 / (r_m * r_m)) + -1.5;
	} else {
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 8d+18) then
        tmp = (2.0d0 / (r_m * r_m)) + (-1.5d0)
    else
        tmp = (r_m * r_m) * ((-0.25d0) * (w * w))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 8e+18) {
		tmp = (2.0 / (r_m * r_m)) + -1.5;
	} else {
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 8e+18:
		tmp = (2.0 / (r_m * r_m)) + -1.5
	else:
		tmp = (r_m * r_m) * (-0.25 * (w * w))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 8e+18)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + -1.5);
	else
		tmp = Float64(Float64(r_m * r_m) * Float64(-0.25 * Float64(w * w)));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 8e+18)
		tmp = (2.0 / (r_m * r_m)) + -1.5;
	else
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 8e+18], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 8e18

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   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 64.8%

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]

   if 8e18 < r

   1. Initial program 91.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.2%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in v around inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \frac{-3}{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{-1}{4} \cdot \left(\color{blue}{{r}^{2}} \cdot {w}^{2}\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left({r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left({w}^{2} \cdot \frac{-1}{4}\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{{w}^{2}} \cdot \frac{-1}{4}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-1}{4}}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right)\right) \]

      14. *-lowering-*.f64 75.3%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right)\right) \]

   8. Simplified 75.3%

      \[\leadsto \color{blue}{-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} \]

   9. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   10. 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 61.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right) \]

   11. Simplified 61.8%

       \[\leadsto \color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.25\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 72.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(r\_m \cdot r\_m\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 3.7e-12) (/ (/ 2.0 r_m) r_m) (* (* r_m r_m) (* -0.375 (* w w)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.7e-12) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = (r_m * r_m) * (-0.375 * (w * w));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 3.7d-12) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (r_m * r_m) * ((-0.375d0) * (w * w))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.7e-12) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = (r_m * r_m) * (-0.375 * (w * w));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 3.7e-12:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = (r_m * r_m) * (-0.375 * (w * w))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 3.7e-12)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(Float64(r_m * r_m) * Float64(-0.375 * Float64(w * w)));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 3.7e-12)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = (r_m * r_m) * (-0.375 * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 3.7e-12], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.69999999999999999e-12

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 3.69999999999999999e-12 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]

   8. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-3}{8}} \]

      2. associate-*l* N/A

         \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} \]

      3. *-commutative N/A

         \[\leadsto {r}^{2} \cdot \left(\frac{-3}{8} \cdot \color{blue}{{w}^{2}}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\frac{-3}{8}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 64.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 64.8%

       \[\leadsto \color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 72.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(r\_m \cdot r\_m\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 4.5e-11) (/ (/ 2.0 r_m) r_m) (* (* r_m r_m) (* w (* w -0.375)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4.5e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = (r_m * r_m) * (w * (w * -0.375));
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 4.5d-11) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = (r_m * r_m) * (w * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4.5e-11) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = (r_m * r_m) * (w * (w * -0.375));
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 4.5e-11:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = (r_m * r_m) * (w * (w * -0.375))
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 4.5e-11)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = Float64(Float64(r_m * r_m) * Float64(w * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 4.5e-11)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = (r_m * r_m) * (w * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 4.5e-11], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(w * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.5e-11

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 4.5e-11 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)}, \frac{-3}{2}\right)\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left({r}^{2} \cdot {w}^{2}\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left({r}^{2} \cdot \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({r}^{2}\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

      8. *-lowering-*.f64 79.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right), \frac{-3}{2}\right)\right) \]

   7. Simplified 79.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)} + -1.5\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      2. swap-sqr N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(\left(w \cdot w\right) \cdot r\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(w \cdot r\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left(w \cdot \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

      11. *-lowering-*.f64 89.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), \frac{-3}{8}\right), \frac{-3}{2}\right)\right) \]

   9. Applied egg-rr 89.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375} + -1.5\right) \]

   10. Taylor expanded in r around inf

       \[\leadsto \color{blue}{\frac{-3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({r}^{2} \cdot {w}^{2}\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. associate-*l* N/A

          \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \frac{-3}{8}\right)} \]

       3. *-commutative N/A

          \[\leadsto {r}^{2} \cdot \left(\frac{-3}{8} \cdot \color{blue}{{w}^{2}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({r}^{2}\right), \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2}\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(r \cdot r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\color{blue}{\frac{-3}{8}} \cdot {w}^{2}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left({w}^{2} \cdot \color{blue}{\frac{-3}{8}}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(\left(w \cdot w\right) \cdot \frac{-3}{8}\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \left(w \cdot \color{blue}{\left(w \cdot \frac{-3}{8}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \color{blue}{\left(w \cdot \frac{-3}{8}\right)}\right)\right) \]

       11. *-lowering-*.f64 64.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(w, \color{blue}{\frac{-3}{8}}\right)\right)\right) \]

   12. Simplified 64.8%

       \[\leadsto \color{blue}{\left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 55.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.4e-10) (/ (/ 2.0 r_m) r_m) -1.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.4d-10) then
        tmp = (2.0d0 / r_m) / r_m
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = (2.0 / r_m) / r_m;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.4e-10:
		tmp = (2.0 / r_m) / r_m
	else:
		tmp = -1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.4e-10)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.4e-10)
		tmp = (2.0 / r_m) / r_m;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.4e-10], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.40000000000000008e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 59.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   9. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} \]

   if 1.40000000000000008e-10 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 20.4%

      \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 55.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.4e-10) (/ 2.0 (* r_m r_m)) -1.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = 2.0 / (r_m * r_m);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.4d-10) then
        tmp = 2.0d0 / (r_m * r_m)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.4e-10) {
		tmp = 2.0 / (r_m * r_m);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.4e-10:
		tmp = 2.0 / (r_m * r_m)
	else:
		tmp = -1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.4e-10)
		tmp = Float64(2.0 / Float64(r_m * r_m));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.4e-10)
		tmp = 2.0 / (r_m * r_m);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.4e-10], N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.40000000000000008e-10

   1. Initial program 86.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 59.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   7. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

   if 1.40000000000000008e-10 < r

   1. Initial program 91.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   4. Step-by-step derivation

   5. Simplified 91.3%

      \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

   6. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 20.4%

      \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 56.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \frac{2}{r\_m \cdot r\_m} + -1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (+ (/ 2.0 (* r_m r_m)) -1.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) + -1.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = (2.0d0 / (r_m * r_m)) + (-1.5d0)
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (2.0 / (r_m * r_m)) + -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (2.0 / (r_m * r_m)) + -1.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(2.0 / Float64(r_m * r_m)) + -1.5)
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = (2.0 / (r_m * r_m)) + -1.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 87.5%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Step-by-step derivation

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
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 52.2%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
8. Add Preprocessing

Alternative 24: 14.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 -1.5)

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -1.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = -1.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return -1.5
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = -1.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := -1.5

Derivation

1. Initial program 87.5%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in r around inf

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\color{blue}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), r\right), r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
4. Step-by-step derivation

5. Simplified 51.6%

   \[\leadsto \left(\color{blue}{3} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]

6. Taylor expanded in w around 0

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
7. Step-by-step derivation

8. Simplified 10.5%

   \[\leadsto \color{blue}{-1.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))