Rosa's TurbineBenchmark

Percentage Accurate: 85.4% → 99.3%

Time: 12.9s

Alternatives: 10

Speedup: 1.1×

• 53.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r r)))
  (+ (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ w (/ (- 1.0 v) r)))) 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))) * ((r * w) * (w / ((1.0 - v) / r)))) + 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))) * ((r * w) * (w / ((1.0d0 - v) / r)))) + 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))) * ((r * w) * (w / ((1.0 - v) / r)))) + 4.5);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.5%

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

   1. associate-/l* 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*r/ 89.2%

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

   4. associate-*l* 97.0%

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

   5. associate-*r* 99.8%

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

   6. clear-num 99.8%

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

   7. un-div-inv 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;\left(3 + t\_0\right) - \left(4.5 + \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 10^{+139}:\\ \;\;\;\;t\_0 + \left(-1.5 - \left(v \cdot \left(-0.25 + \frac{0.375}{v}\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.4e-51)
     (- (+ 3.0 t_0) (+ 4.5 (/ (* r (* w 0.375)) (/ (- 1.0 v) (* r w)))))
     (if (<= r 1e+139)
       (+
        t_0
        (-
         -1.5
         (* (* v (+ -0.25 (/ 0.375 v))) (* r (* (* w w) (/ r (- 1.0 v)))))))
       (-
        3.0
        (+
         4.5
         (/ (* (* r w) (* w (+ 0.375 (* v -0.25)))) (/ (- 1.0 v) r))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.4e-51) {
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	} else if (r <= 1e+139) {
		tmp = t_0 + (-1.5 - ((v * (-0.25 + (0.375 / v))) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.4d-51) then
        tmp = (3.0d0 + t_0) - (4.5d0 + ((r * (w * 0.375d0)) / ((1.0d0 - v) / (r * w))))
    else if (r <= 1d+139) then
        tmp = t_0 + ((-1.5d0) - ((v * ((-0.25d0) + (0.375d0 / v))) * (r * ((w * w) * (r / (1.0d0 - v))))))
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (w * (0.375d0 + (v * (-0.25d0))))) / ((1.0d0 - v) / r)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.4e-51) {
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	} else if (r <= 1e+139) {
		tmp = t_0 + (-1.5 - ((v * (-0.25 + (0.375 / v))) * (r * ((w * w) * (r / (1.0 - v))))));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.4e-51:
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))))
	elif r <= 1e+139:
		tmp = t_0 + (-1.5 - ((v * (-0.25 + (0.375 / v))) * (r * ((w * w) * (r / (1.0 - v))))))
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.4e-51)
		tmp = Float64(Float64(3.0 + t_0) - Float64(4.5 + Float64(Float64(r * Float64(w * 0.375)) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	elseif (r <= 1e+139)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(v * Float64(-0.25 + Float64(0.375 / v))) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(1.0 - v)))))));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(w * Float64(0.375 + Float64(v * -0.25)))) / Float64(Float64(1.0 - v) / r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.4e-51)
		tmp = (3.0 + t_0) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	elseif (r <= 1e+139)
		tmp = t_0 + (-1.5 - ((v * (-0.25 + (0.375 / v))) * (r * ((w * w) * (r / (1.0 - v))))));
	else
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.4e-51], N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(4.5 + N[(N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1e+139], N[(t$95$0 + N[(-1.5 - N[(N[(v * N[(-0.25 + N[(0.375 / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.4e-51

   1. Initial program 86.0%

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

   3. Simplified 87.6%

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

      1. associate-/l* 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*r/ 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-*l* 97.2%

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

      6. associate-*l* 98.8%

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

      7. clear-num 98.8%

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

      8. un-div-inv 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*r* 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 98.8%

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

      5. *-commutative 98.8%

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

      6. clear-num 98.7%

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

      7. un-div-inv 98.8%

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

      8. distribute-rgt-in 98.8%

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

      9. metadata-eval 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-*l* 98.8%

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

      12. metadata-eval 98.8%

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

      13. associate-/l/ 98.7%

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

      14. *-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in v around 0 88.6%

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

       1. *-commutative 88.6%

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

       2. *-commutative 88.6%

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

       3. *-commutative 88.6%

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

       4. associate-*l* 88.6%

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

   11. Simplified 88.6%

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

   if 2.4e-51 < r < 1.00000000000000003e139

   1. Initial program 91.6%

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

   3. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 95.7%

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

      1. sub-neg 95.7%

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

      2. associate-*r/ 95.8%

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

      3. metadata-eval 95.8%

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

      4. metadata-eval 95.8%

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

   7. Simplified 95.8%

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

   if 1.00000000000000003e139 < r

   1. Initial program 86.2%

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

   3. Simplified 91.7%

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

      1. associate-/l* 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*r/ 91.8%

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

      4. *-commutative 91.8%

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

      5. associate-*l* 94.1%

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

      6. associate-*l* 97.0%

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

      7. clear-num 96.9%

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

      8. un-div-inv 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. associate-*r* 96.8%

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

      2. associate-*l/ 96.9%

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

      3. *-commutative 96.9%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*l* 99.8%

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

      10. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.4 \cdot 10^{-51}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{elif}\;r \leq 10^{+139}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot \left(-0.25 + \frac{0.375}{v}\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 10^{-50}:\\ \;\;\;\;t\_0 - \left(4.5 + \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (<= r 1e-50)
     (- t_0 (+ 4.5 (/ (* r (* w 0.375)) (/ (- 1.0 v) (* r w)))))
     (-
      t_0
      (+ 4.5 (/ (* (* r w) (* w (+ 0.375 (* v -0.25)))) (/ (- 1.0 v) r)))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (r <= 1e-50) {
		tmp = t_0 - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	} else {
		tmp = t_0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if (r <= 1d-50) then
        tmp = t_0 - (4.5d0 + ((r * (w * 0.375d0)) / ((1.0d0 - v) / (r * w))))
    else
        tmp = t_0 - (4.5d0 + (((r * w) * (w * (0.375d0 + (v * (-0.25d0))))) / ((1.0d0 - v) / r)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if (r <= 1e-50) {
		tmp = t_0 - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	} else {
		tmp = t_0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if r <= 1e-50:
		tmp = t_0 - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))))
	else:
		tmp = t_0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 1e-50)
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(r * Float64(w * 0.375)) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	else
		tmp = Float64(t_0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(w * Float64(0.375 + Float64(v * -0.25)))) / Float64(Float64(1.0 - v) / r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if (r <= 1e-50)
		tmp = t_0 - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	else
		tmp = t_0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1e-50], N[(t$95$0 - N[(4.5 + N[(N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 1.00000000000000001e-50

   1. Initial program 86.0%

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

   3. Simplified 87.6%

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

      1. associate-/l* 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*r/ 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-*l* 97.2%

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

      6. associate-*l* 98.8%

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

      7. clear-num 98.8%

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

      8. un-div-inv 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*r* 91.6%

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

      2. *-commutative 91.6%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 98.8%

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

      5. *-commutative 98.8%

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

      6. clear-num 98.7%

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

      7. un-div-inv 98.8%

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

      8. distribute-rgt-in 98.8%

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

      9. metadata-eval 98.8%

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

      10. *-commutative 98.8%

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

      11. associate-*l* 98.8%

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

      12. metadata-eval 98.8%

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

      13. associate-/l/ 98.7%

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

      14. *-commutative 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in v around 0 88.6%

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

       1. *-commutative 88.6%

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

       2. *-commutative 88.6%

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

       3. *-commutative 88.6%

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

       4. associate-*l* 88.6%

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

   11. Simplified 88.6%

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

   if 1.00000000000000001e-50 < r

   1. Initial program 89.3%

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

   3. Simplified 94.1%

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

      1. associate-/l* 94.1%

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

      2. *-commutative 94.1%

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

      3. associate-*r/ 94.1%

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

      4. *-commutative 94.1%

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

      5. associate-*l* 96.3%

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

      6. associate-*l* 98.6%

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

      7. clear-num 98.5%

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

      8. un-div-inv 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. associate-*r* 97.3%

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

      2. associate-*l/ 97.3%

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

      3. *-commutative 97.3%

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

      4. associate-*r/ 96.4%

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

      5. *-commutative 96.4%

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

      6. distribute-rgt-in 96.4%

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

      7. metadata-eval 96.4%

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

      8. *-commutative 96.4%

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

      9. associate-*l* 96.4%

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

      10. metadata-eval 96.4%

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

   8. Applied egg-rr 96.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{-50}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.75 \cdot 10^{-148}:\\ \;\;\;\;t\_1 + \left(-1.5 - \left(v \cdot -0.25\right) \cdot t\_0\right)\\ \mathbf{elif}\;r \leq 4 \cdot 10^{+44}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r (* w w)))) (t_1 (/ 2.0 (* r r))))
   (if (<= r 2.75e-148)
     (+ t_1 (- -1.5 (* (* v -0.25) t_0)))
     (if (<= r 4e+44)
       (+ t_1 (- -1.5 (* 0.375 t_0)))
       (-
        3.0
        (+
         4.5
         (/ (* (* r w) (* w (+ 0.375 (* v -0.25)))) (/ (- 1.0 v) r))))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.75e-148) {
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0));
	} else if (r <= 4e+44) {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * (r * (w * w))
    t_1 = 2.0d0 / (r * r)
    if (r <= 2.75d-148) then
        tmp = t_1 + ((-1.5d0) - ((v * (-0.25d0)) * t_0))
    else if (r <= 4d+44) then
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (w * (0.375d0 + (v * (-0.25d0))))) / ((1.0d0 - v) / r)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.75e-148) {
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0));
	} else if (r <= 4e+44) {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * (w * w))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.75e-148:
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0))
	elif r <= 4e+44:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * Float64(w * w)))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.75e-148)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(v * -0.25) * t_0)));
	elseif (r <= 4e+44)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(w * Float64(0.375 + Float64(v * -0.25)))) / Float64(Float64(1.0 - v) / r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * (w * w));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.75e-148)
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0));
	elseif (r <= 4e+44)
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	else
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.75e-148], N[(t$95$1 + N[(-1.5 - N[(N[(v * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 4e+44], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.7500000000000001e-148

   1. Initial program 86.4%

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

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in v around 0 69.2%

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

   if 2.7500000000000001e-148 < r < 4.0000000000000004e44

   1. Initial program 86.2%

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

   3. Simplified 88.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 79.3%

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

   6. Taylor expanded in v around 0 86.1%

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

   if 4.0000000000000004e44 < r

   1. Initial program 89.3%

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

   3. Simplified 94.7%

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

      1. associate-/l* 94.7%

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

      2. *-commutative 94.7%

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

      3. associate-*r/ 94.7%

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

      4. *-commutative 94.7%

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

      5. associate-*l* 96.2%

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

      6. associate-*l* 98.0%

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

      7. clear-num 98.0%

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

      8. un-div-inv 98.0%

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

   6. Applied egg-rr 98.0%

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

      1. associate-*r* 97.9%

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

      2. associate-*l/ 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*l* 99.8%

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

      10. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.75 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{elif}\;r \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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))) - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * (w * (r * (w / ((1.0d0 - v) / r))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.5%

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

   1. associate-/l* 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*r/ 89.2%

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

   4. *-commutative 89.2%

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

   5. associate-*l* 97.0%

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

   6. associate-*l* 98.7%

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

   7. clear-num 98.7%

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

   8. un-div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 6: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3800000:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3800000.0)
   (-
    (+ 3.0 (/ 2.0 (* r r)))
    (+ 4.5 (/ (* r (* w 0.375)) (/ (- 1.0 v) (* r w)))))
   (-
    3.0
    (+ 4.5 (/ (* (* r w) (* w (+ 0.375 (* v -0.25)))) (/ (- 1.0 v) r))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3800000.0) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 3800000.0d0) then
        tmp = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + ((r * (w * 0.375d0)) / ((1.0d0 - v) / (r * w))))
    else
        tmp = 3.0d0 - (4.5d0 + (((r * w) * (w * (0.375d0 + (v * (-0.25d0))))) / ((1.0d0 - v) / r)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 3800000.0) {
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	} else {
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3800000.0:
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))))
	else:
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3800000.0)
		tmp = Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(Float64(r * Float64(w * 0.375)) / Float64(Float64(1.0 - v) / Float64(r * w)))));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(w * Float64(0.375 + Float64(v * -0.25)))) / Float64(Float64(1.0 - v) / r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3800000.0)
		tmp = (3.0 + (2.0 / (r * r))) - (4.5 + ((r * (w * 0.375)) / ((1.0 - v) / (r * w))));
	else
		tmp = 3.0 - (4.5 + (((r * w) * (w * (0.375 + (v * -0.25)))) / ((1.0 - v) / r)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3800000.0], N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.5 + N[(N[(r * N[(w * 0.375), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(N[(r * w), $MachinePrecision] * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.8e6

   1. Initial program 86.1%

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

   3. Simplified 87.5%

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

      1. associate-/l* 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*r/ 87.1%

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

      4. *-commutative 87.1%

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

      5. associate-*l* 97.0%

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

      6. associate-*l* 98.8%

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

      7. clear-num 98.8%

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

      8. un-div-inv 98.8%

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

   6. Applied egg-rr 98.8%

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

      1. associate-*r* 91.7%

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

      2. *-commutative 91.7%

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

      3. associate-*r* 92.7%

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

      4. associate-*r* 98.3%

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

      5. *-commutative 98.3%

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

      6. clear-num 98.3%

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

      7. un-div-inv 98.4%

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

      8. distribute-rgt-in 98.4%

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

      9. metadata-eval 98.4%

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

      10. *-commutative 98.4%

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

      11. associate-*l* 98.4%

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

      12. metadata-eval 98.4%

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

      13. associate-/l/ 98.3%

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

      14. *-commutative 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in v around 0 87.3%

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

       1. *-commutative 87.3%

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

       2. *-commutative 87.3%

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

       3. *-commutative 87.3%

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

       4. associate-*l* 87.4%

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

   11. Simplified 87.4%

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

   if 3.8e6 < r

   1. Initial program 89.8%

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

   3. Simplified 95.6%

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

      1. associate-/l* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r/ 95.7%

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

      4. *-commutative 95.7%

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

      5. associate-*l* 96.9%

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

      6. associate-*l* 98.4%

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

      7. clear-num 98.3%

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

      8. un-div-inv 98.3%

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

   6. Applied egg-rr 98.3%

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

      1. associate-*r* 98.3%

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

      2. associate-*l/ 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r/ 98.4%

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

      5. *-commutative 98.4%

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

      6. distribute-rgt-in 98.4%

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

      7. metadata-eval 98.4%

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

      8. *-commutative 98.4%

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

      9. associate-*l* 98.4%

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

      10. metadata-eval 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in r around inf 98.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3800000:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{r \cdot \left(w \cdot 0.375\right)}{\frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)}{\frac{1 - v}{r}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{1 - v}{r \cdot w}}\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r r)))
  (+ 4.5 (/ (* (* r w) (+ 0.375 (* v -0.25))) (/ (- 1.0 v) (* r w))))))

C

double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (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 = (3.0d0 + (2.0d0 / (r * r))) - (4.5d0 + (((r * w) * (0.375d0 + (v * (-0.25d0)))) / ((1.0d0 - v) / (r * w))))
end function

Java

public static double code(double v, double w, double r) {
	return (3.0 + (2.0 / (r * r))) - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (r * w))));
}

Python

def code(v, w, r):
	return (3.0 + (2.0 / (r * r))) - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (r * w))))

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(4.5 + Float64(Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))) / Float64(Float64(1.0 - v) / Float64(r * w)))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (((r * w) * (0.375 + (v * -0.25))) / ((1.0 - v) / (r * w))));
end

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.5%

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

   1. associate-/l* 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*r/ 89.2%

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

   4. *-commutative 89.2%

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

   5. associate-*l* 97.0%

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

   6. associate-*l* 98.7%

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

   7. clear-num 98.7%

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

   8. un-div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

   1. associate-*r* 93.3%

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

   2. *-commutative 93.3%

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

   3. associate-*r* 93.4%

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

   4. associate-*r* 97.6%

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

   5. *-commutative 97.6%

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

   6. clear-num 97.6%

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

   7. un-div-inv 97.7%

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

   8. distribute-rgt-in 97.7%

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

   9. metadata-eval 97.7%

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

   10. *-commutative 97.7%

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

   11. associate-*l* 97.7%

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

   12. metadata-eval 97.7%

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

   13. associate-/l/ 97.6%

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

   14. *-commutative 97.6%

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

8. Applied egg-rr 97.6%

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

9. Final simplification 97.6%

   \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + \frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{1 - v}{r \cdot w}}\right) \]
10. Add Preprocessing

Alternative 8: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3 \cdot 10^{-148}:\\ \;\;\;\;t\_1 + \left(-1.5 - \left(v \cdot -0.25\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 - 0.375 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r (* w w)))) (t_1 (/ 2.0 (* r r))))
   (if (<= r 3e-148)
     (+ t_1 (- -1.5 (* (* v -0.25) t_0)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 3e-148) {
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * 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) :: t_1
    real(8) :: tmp
    t_0 = r * (r * (w * w))
    t_1 = 2.0d0 / (r * r)
    if (r <= 3d-148) then
        tmp = t_1 + ((-1.5d0) - ((v * (-0.25d0)) * t_0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * (r * (w * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 3e-148) {
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * (w * w))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if r <= 3e-148:
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * Float64(w * w)))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3e-148)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(v * -0.25) * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * (w * w));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 3e-148)
		tmp = t_1 + (-1.5 - ((v * -0.25) * t_0));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3e-148], N[(t$95$1 + N[(-1.5 - N[(N[(v * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.99999999999999998e-148

   1. Initial program 86.4%

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

   3. Simplified 87.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in v around 0 69.2%

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

   if 2.99999999999999998e-148 < r

   1. Initial program 87.8%

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

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 71.4%

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

   6. Taylor expanded in v around 0 87.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + \frac{\frac{2}{r}}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.7e-94)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* 0.375 (* r (* (* w w) (/ r v))))))
   (+ (- -1.5 (* 0.375 (* r (* r (* w w))))) (/ (/ 2.0 r) r))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.7e-94) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / v)))));
	} else {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) + ((2.0 / r) / r);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 4.7d-94) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / v)))))
    else
        tmp = ((-1.5d0) - (0.375d0 * (r * (r * (w * w))))) + ((2.0d0 / r) / r)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.7e-94) {
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / v)))));
	} else {
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) + ((2.0 / r) / r);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4.7e-94:
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / v)))))
	else:
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) + ((2.0 / r) / r)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4.7e-94)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / v))))));
	else
		tmp = Float64(Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))) + Float64(Float64(2.0 / r) / r));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4.7e-94)
		tmp = (2.0 / (r * r)) + (-1.5 + (0.375 * (r * ((w * w) * (r / v)))));
	else
		tmp = (-1.5 - (0.375 * (r * (r * (w * w))))) + ((2.0 / r) / r);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.7e-94], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.5 - N[(0.375 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.70000000000000003e-94

   1. Initial program 87.3%

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

   3. Simplified 88.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 79.5%

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

   6. Taylor expanded in v around inf 69.5%

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

      1. neg-mul-1 69.5%

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

   8. Simplified 69.5%

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

   if 4.70000000000000003e-94 < r

   1. Initial program 86.3%

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

   3. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 67.8%

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

   6. Taylor expanded in v around 0 85.8%

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

      1. associate-/r* 85.9%

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

      2. div-inv 85.8%

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

   8. Applied egg-rr 85.8%

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

      1. associate-*r/ 85.9%

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

      2. *-rgt-identity 85.9%

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

   10. Simplified 85.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) + \frac{\frac{2}{r}}{r}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right) \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 - (0.375 * (r * (r * (w * 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) - (0.375d0 * (r * (r * (w * w)))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 - (0.375 * (r * (r * (w * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 - (0.375 * (r * (r * (w * w)))))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(0.375 * Float64(r * Float64(r * Float64(w * w))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Simplified 89.2%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in v around 0 75.5%

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

6. Taylor expanded in v around 0 85.6%

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

7. Final simplification 85.6%

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

Reproduce

herbie shell --seed 2024113 
(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))