Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.4%

Time: 15.6s

Alternatives: 13

Speedup: 1.7×

• 52.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r} - \left(1.5 + \left(r \cdot \left(w \cdot 0.25\right)\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{if}\;v \leq -1.65 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;v \leq 0.6:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (- (/ (/ 2.0 r) r) (+ 1.5 (* (* r (* w 0.25)) (* r w))))))
   (if (<= v -1.65e+81)
     t_0
     (if (<= v 0.6)
       (-
        (+
         (+ 3.0 (/ 2.0 (* r r)))
         (/ (* (* r w) (* r (* w (+ 0.375 (* v -0.25))))) (+ v -1.0)))
        4.5)
       t_0))))

C

double code(double v, double w, double r) {
	double t_0 = ((2.0 / r) / r) - (1.5 + ((r * (w * 0.25)) * (r * w)));
	double tmp;
	if (v <= -1.65e+81) {
		tmp = t_0;
	} else if (v <= 0.6) {
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.0d0 / r) / r) - (1.5d0 + ((r * (w * 0.25d0)) * (r * w)))
    if (v <= (-1.65d+81)) then
        tmp = t_0
    else if (v <= 0.6d0) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) + (((r * w) * (r * (w * (0.375d0 + (v * (-0.25d0)))))) / (v + (-1.0d0)))) - 4.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = ((2.0 / r) / r) - (1.5 + ((r * (w * 0.25)) * (r * w)));
	double tmp;
	if (v <= -1.65e+81) {
		tmp = t_0;
	} else if (v <= 0.6) {
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = ((2.0 / r) / r) - (1.5 + ((r * (w * 0.25)) * (r * w)))
	tmp = 0
	if v <= -1.65e+81:
		tmp = t_0
	elif v <= 0.6:
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5
	else:
		tmp = t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(Float64(2.0 / r) / r) - Float64(1.5 + Float64(Float64(r * Float64(w * 0.25)) * Float64(r * w))))
	tmp = 0.0
	if (v <= -1.65e+81)
		tmp = t_0;
	elseif (v <= 0.6)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(Float64(r * w) * Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25))))) / Float64(v + -1.0))) - 4.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = ((2.0 / r) / r) - (1.5 + ((r * (w * 0.25)) * (r * w)));
	tmp = 0.0;
	if (v <= -1.65e+81)
		tmp = t_0;
	elseif (v <= 0.6)
		tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision] - N[(1.5 + N[(N[(r * N[(w * 0.25), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -1.65e+81], t$95$0, If[LessEqual[v, 0.6], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.65e81 or 0.599999999999999978 < v

   1. Initial program 82.5%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 82.0%

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

   5. Simplified 82.0%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 82.0%

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

   7. Applied egg-rr 82.0%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

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

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

      5. associate-*r* N/A

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

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

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

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

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

      8. *-lowering-*.f64 92.4%

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

   9. Applied egg-rr 92.4%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f64 99.9%

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

   11. Applied egg-rr 99.9%

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

   if -1.65e81 < v < 0.599999999999999978

   1. Initial program 85.1%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \left(1.5 + \left(r \cdot \left(w \cdot 0.25\right)\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{elif}\;v \leq 0.6:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \left(1.5 + \left(r \cdot \left(w \cdot 0.25\right)\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 0.375 + v \cdot -0.25\\ \mathbf{if}\;r \leq 8.5 \cdot 10^{-71}:\\ \;\;\;\;t\_0 - \left(1.5 + \left(w \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;r \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\left(\left(3 + t\_0\right) - \left(r \cdot t\_1\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot t\_1\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 0.375 (* v -0.25))))
   (if (<= r 8.5e-71)
     (- t_0 (+ 1.5 (* (* w 0.25) (* r (* r w)))))
     (if (<= r 2e+115)
       (- (- (+ 3.0 t_0) (* (* r t_1) (/ (* r (* w w)) (- 1.0 v)))) 4.5)
       (- (+ 3.0 (/ (* (* r w) (* r (* w t_1))) (+ v -1.0))) 4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.375 + (v * -0.25);
	double tmp;
	if (r <= 8.5e-71) {
		tmp = t_0 - (1.5 + ((w * 0.25) * (r * (r * w))));
	} else if (r <= 2e+115) {
		tmp = ((3.0 + t_0) - ((r * t_1) * ((r * (w * w)) / (1.0 - v)))) - 4.5;
	} else {
		tmp = (3.0 + (((r * w) * (r * (w * t_1))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 0.375d0 + (v * (-0.25d0))
    if (r <= 8.5d-71) then
        tmp = t_0 - (1.5d0 + ((w * 0.25d0) * (r * (r * w))))
    else if (r <= 2d+115) then
        tmp = ((3.0d0 + t_0) - ((r * t_1) * ((r * (w * w)) / (1.0d0 - v)))) - 4.5d0
    else
        tmp = (3.0d0 + (((r * w) * (r * (w * t_1))) / (v + (-1.0d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 0.375 + (v * -0.25);
	double tmp;
	if (r <= 8.5e-71) {
		tmp = t_0 - (1.5 + ((w * 0.25) * (r * (r * w))));
	} else if (r <= 2e+115) {
		tmp = ((3.0 + t_0) - ((r * t_1) * ((r * (w * w)) / (1.0 - v)))) - 4.5;
	} else {
		tmp = (3.0 + (((r * w) * (r * (w * t_1))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 0.375 + (v * -0.25)
	tmp = 0
	if r <= 8.5e-71:
		tmp = t_0 - (1.5 + ((w * 0.25) * (r * (r * w))))
	elif r <= 2e+115:
		tmp = ((3.0 + t_0) - ((r * t_1) * ((r * (w * w)) / (1.0 - v)))) - 4.5
	else:
		tmp = (3.0 + (((r * w) * (r * (w * t_1))) / (v + -1.0))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(0.375 + Float64(v * -0.25))
	tmp = 0.0
	if (r <= 8.5e-71)
		tmp = Float64(t_0 - Float64(1.5 + Float64(Float64(w * 0.25) * Float64(r * Float64(r * w)))));
	elseif (r <= 2e+115)
		tmp = Float64(Float64(Float64(3.0 + t_0) - Float64(Float64(r * t_1) * Float64(Float64(r * Float64(w * w)) / Float64(1.0 - v)))) - 4.5);
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(r * w) * Float64(r * Float64(w * t_1))) / Float64(v + -1.0))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 0.375 + (v * -0.25);
	tmp = 0.0;
	if (r <= 8.5e-71)
		tmp = t_0 - (1.5 + ((w * 0.25) * (r * (r * w))));
	elseif (r <= 2e+115)
		tmp = ((3.0 + t_0) - ((r * t_1) * ((r * (w * w)) / (1.0 - v)))) - 4.5;
	else
		tmp = (3.0 + (((r * w) * (r * (w * t_1))) / (v + -1.0))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 8.5e-71], N[(t$95$0 - N[(1.5 + N[(N[(w * 0.25), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 2e+115], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(r * t$95$1), $MachinePrecision] * N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 8.49999999999999988e-71

   1. Initial program 80.5%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 73.2%

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

   5. Simplified 73.2%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 93.8%

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

   7. Applied egg-rr 93.8%

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

   if 8.49999999999999988e-71 < r < 2e115

   1. Initial program 95.3%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 95.3%

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

   4. Applied egg-rr 95.3%

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

      1. associate-/l* N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. associate-/l* N/A

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

      9. associate-*r* N/A

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

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

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

   6. Applied egg-rr 99.6%

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

   if 2e115 < r

   1. Initial program 87.4%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 97.6%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in r around inf

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

   9. Simplified 97.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + \left(w \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;r \leq 2 \cdot 10^{+115}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \frac{r \cdot \left(w \cdot w\right)}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + \left(w \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.9e+25)
   (- (/ 2.0 (* r r)) (+ 1.5 (* (* w 0.25) (* r (* r w)))))
   (-
    (+ 3.0 (/ (* (* r w) (* r (* w (+ 0.375 (* v -0.25))))) (+ v -1.0)))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.9e+25) {
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))));
	} else {
		tmp = (3.0 + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.9d+25) then
        tmp = (2.0d0 / (r * r)) - (1.5d0 + ((w * 0.25d0) * (r * (r * w))))
    else
        tmp = (3.0d0 + (((r * w) * (r * (w * (0.375d0 + (v * (-0.25d0)))))) / (v + (-1.0d0)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.9e+25) {
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))));
	} else {
		tmp = (3.0 + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.9e+25:
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))))
	else:
		tmp = (3.0 + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.9e+25)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - Float64(1.5 + Float64(Float64(w * 0.25) * Float64(r * Float64(r * w)))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(r * w) * Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25))))) / Float64(v + -1.0))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.9e+25)
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))));
	else
		tmp = (3.0 + (((r * w) * (r * (w * (0.375 + (v * -0.25))))) / (v + -1.0))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.9e+25], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(1.5 + N[(N[(w * 0.25), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.8999999999999999e25

   1. Initial program 82.0%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 75.7%

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

   5. Simplified 75.7%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 93.7%

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

   7. Applied egg-rr 93.7%

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

   if 2.8999999999999999e25 < r

   1. Initial program 90.9%

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

      1. associate-*l* N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-in N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

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

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

      18. metadata-eval N/A

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

      19. *-lowering-*.f64 98.3%

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in r around inf

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

   9. Simplified 98.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + \left(w \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7.5:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + \left(w \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7.5)
   (- (/ 2.0 (* r r)) (+ 1.5 (* (* w 0.25) (* r (* r w)))))
   (-
    (+ 3.0 (* (+ 0.375 (* v -0.25)) (/ (* r (* r (* w w))) (+ v -1.0))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.5) {
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))));
	} else {
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7.5d0) then
        tmp = (2.0d0 / (r * r)) - (1.5d0 + ((w * 0.25d0) * (r * (r * w))))
    else
        tmp = (3.0d0 + ((0.375d0 + (v * (-0.25d0))) * ((r * (r * (w * w))) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.5) {
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))));
	} else {
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7.5:
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))))
	else:
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7.5)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - Float64(1.5 + Float64(Float64(w * 0.25) * Float64(r * Float64(r * w)))));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * Float64(r * Float64(w * w))) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7.5)
		tmp = (2.0 / (r * r)) - (1.5 + ((w * 0.25) * (r * (r * w))));
	else
		tmp = (3.0 + ((0.375 + (v * -0.25)) * ((r * (r * (w * w))) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7.5], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(1.5 + N[(N[(w * 0.25), $MachinePrecision] * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(r * N[(w * 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 < 7.5

   1. Initial program 82.4%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 75.5%

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

   5. Simplified 75.5%

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

      1. associate-*l* N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-lowering-*.f64 93.9%

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

   7. Applied egg-rr 93.9%

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

   if 7.5 < r

   1. Initial program 88.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. sub-neg N/A

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

      12. distribute-lft-in N/A

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

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

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

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

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

      19. metadata-eval 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in r around inf

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

   7. Simplified 93.1%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. *-lowering-*.f64 93.1%

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

   9. Applied egg-rr 93.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.5:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + \left(w \cdot 0.25\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 150000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + w \cdot \left(\left(w \cdot 0.25\right) \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 150000000.0)
   (- (/ 2.0 (* r r)) (+ 1.5 (* w (* (* w 0.25) (* r r)))))
   (+ -1.5 (* r (* -0.25 (* r (* w w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 150000000.0) {
		tmp = (2.0 / (r * r)) - (1.5 + (w * ((w * 0.25) * (r * r))));
	} else {
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 150000000.0d0) then
        tmp = (2.0d0 / (r * r)) - (1.5d0 + (w * ((w * 0.25d0) * (r * r))))
    else
        tmp = (-1.5d0) + (r * ((-0.25d0) * (r * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 150000000.0) {
		tmp = (2.0 / (r * r)) - (1.5 + (w * ((w * 0.25) * (r * r))));
	} else {
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 150000000.0:
		tmp = (2.0 / (r * r)) - (1.5 + (w * ((w * 0.25) * (r * r))))
	else:
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 150000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) - Float64(1.5 + Float64(w * Float64(Float64(w * 0.25) * Float64(r * r)))));
	else
		tmp = Float64(-1.5 + Float64(r * Float64(-0.25 * Float64(r * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 150000000.0)
		tmp = (2.0 / (r * r)) - (1.5 + (w * ((w * 0.25) * (r * r))));
	else
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 150000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] - N[(1.5 + N[(w * N[(N[(w * 0.25), $MachinePrecision] * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r * N[(-0.25 * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.5e8

   1. Initial program 82.5%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 75.6%

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

   5. Simplified 75.6%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 89.1%

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

   7. Applied egg-rr 89.1%

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

   if 1.5e8 < r

   1. Initial program 88.5%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 83.6%

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

   5. Simplified 83.6%

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

   6. Taylor expanded in r around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. neg-mul-1 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. unpow2 N/A

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

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

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

   8. Simplified 83.6%

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

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

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

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

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

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

      7. *-lowering-*.f64 87.6%

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

   10. Applied egg-rr 87.6%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot -0.25\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 150000000:\\ \;\;\;\;\frac{2}{r \cdot r} - \left(1.5 + w \cdot \left(\left(w \cdot 0.25\right) \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in v around inf

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

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

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

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

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

   5. unpow2 N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 77.5%

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

5. Simplified 77.5%

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

   1. associate-/r* N/A

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

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

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

   3. /-lowering-/.f64 77.5%

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

7. Applied egg-rr 77.5%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

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

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

   5. associate-*r* N/A

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

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

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

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

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

   8. *-lowering-*.f64 87.9%

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

9. Applied egg-rr 87.9%

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

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

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

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

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

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

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

    7. *-lowering-*.f64 93.4%

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

11. Applied egg-rr 93.4%

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

12. Final simplification 93.4%

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

Alternative 7: 91.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in v around inf

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

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

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

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

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

   5. unpow2 N/A

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

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

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

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

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 77.5%

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

5. Simplified 77.5%

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

   1. associate-*l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. *-lowering-*.f64 92.5%

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

7. Applied egg-rr 92.5%

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

8. Final simplification 92.5%

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

Alternative 8: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.3:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3.3)
   (+ (/ 2.0 (* r r)) -1.5)
   (+ -1.5 (* r (* -0.25 (* r (* w w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.3) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 3.3d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (-1.5d0) + (r * ((-0.25d0) * (r * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.3) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.3:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.3)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(-1.5 + Float64(r * Float64(-0.25 * Float64(r * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.3)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.3], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + N[(r * N[(-0.25 * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.2999999999999998

   1. Initial program 82.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-commutative N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 70.3%

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

   5. Simplified 70.3%

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

   if 3.2999999999999998 < r

   1. Initial program 88.7%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 83.9%

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

   5. Simplified 83.9%

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

   6. Taylor expanded in r around inf

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

      1. distribute-rgt-in N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. neg-mul-1 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

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

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

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

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

      17. unpow2 N/A

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

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

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

   8. Simplified 83.9%

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

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

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

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

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

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

      7. *-lowering-*.f64 87.8%

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

   10. Applied egg-rr 87.8%

       \[\leadsto -1.5 + \color{blue}{\left(\left(r \cdot \left(w \cdot w\right)\right) \cdot -0.25\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.3:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6.4:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 6.4)
   (+ (/ 2.0 (* r r)) -1.5)
   (+ -1.5 (* (* r r) (* (* w w) -0.375)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 6.4) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -1.5 + ((r * r) * ((w * w) * -0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 6.4d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (-1.5d0) + ((r * r) * ((w * w) * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 6.4) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -1.5 + ((r * r) * ((w * w) * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 6.4:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = -1.5 + ((r * r) * ((w * w) * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 6.4)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(-1.5 + Float64(Float64(r * r) * Float64(Float64(w * w) * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 6.4)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = -1.5 + ((r * r) * ((w * w) * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 6.4], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.4000000000000004

   1. Initial program 82.4%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-commutative N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 70.3%

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

   5. Simplified 70.3%

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

   if 6.4000000000000004 < r

   1. Initial program 88.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

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

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

      11. sub-neg N/A

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

      12. distribute-lft-in N/A

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

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

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

      14. metadata-eval N/A

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

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

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

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

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

      19. metadata-eval 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in r around inf

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

   7. Simplified 93.1%

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

   8. Taylor expanded in v around 0

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

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

      10. unpow2 N/A

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

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

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 82.2%

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

   10. Simplified 82.2%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.4:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e+97) (+ (/ 2.0 (* r r)) -1.5) (* (* r r) (* -0.25 (* w w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e+97) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * (-0.25 * (w * w));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5d+97) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * r) * ((-0.25d0) * (w * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e+97) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * (-0.25 * (w * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5e+97:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * r) * (-0.25 * (w * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e+97)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * r) * Float64(-0.25 * Float64(w * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5e+97)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * r) * (-0.25 * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e+97], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * r), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.99999999999999999e97

   1. Initial program 83.2%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

      3. +-commutative N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 68.6%

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

   5. Simplified 68.6%

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

   if 4.99999999999999999e97 < r

   1. Initial program 87.7%

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

   3. Taylor expanded in v around inf

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

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

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

      11. unpow2 N/A

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

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

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 78.1%

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

   5. Simplified 78.1%

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

   6. Taylor expanded in r around inf

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 71.3%

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

   8. Simplified 71.3%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 82.4%

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

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. 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 56.7%

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

   5. Simplified 56.7%

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

   if 1.1499999999999999 < r

   1. Initial program 88.7%

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

   3. Taylor expanded in r around 0

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

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r \cdot r}\right), \frac{9}{2}\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r}\right), r\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 3 \cdot {r}^{2}\right), r\right), r\right), \frac{9}{2}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(3 \cdot {r}^{2}\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 21.7%

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

   5. Simplified 21.7%

      \[\leadsto \color{blue}{\frac{\frac{2 + \left(r \cdot r\right) \cdot 3}{r}}{r}} - 4.5 \]

   6. Taylor expanded in r around inf

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

   8. Simplified 25.3%

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

Alternative 12: 57.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 59.4%

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

5. Simplified 59.4%

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

6. Final simplification 59.4%

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

Alternative 13: 14.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.0%

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

3. Taylor expanded in r around 0

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

   1. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r \cdot r}\right), \frac{9}{2}\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r}\right), r\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 3 \cdot {r}^{2}\right), r\right), r\right), \frac{9}{2}\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(3 \cdot {r}^{2}\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 55.6%

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

5. Simplified 55.6%

   \[\leadsto \color{blue}{\frac{\frac{2 + \left(r \cdot r\right) \cdot 3}{r}}{r}} - 4.5 \]

6. Taylor expanded in r around inf

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

8. Simplified 16.6%

   \[\leadsto \color{blue}{-1.5} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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))