Rosa's TurbineBenchmark

Percentage Accurate: 85.4% → 99.8%

Time: 15.3s

Alternatives: 5

Speedup: 1.5×

• 53.1% 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: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (3.0 + (pow(r, -2.0) * 2.0)) + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * ((r * w) / (v + -1.0)))) - 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 + ((r ** (-2.0d0)) * 2.0d0)) + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * ((r * w) / (v + (-1.0d0))))) - 4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 86.5%

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

   1. associate-/l* 86.1%

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

   2. *-commutative 86.1%

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

   3. associate-*r/ 85.6%

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

   4. associate-*l* 94.8%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r/ 99.7%

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

6. Applied egg-rr 99.7%

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

   1. clear-num 99.7%

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

   2. associate-/r/ 99.7%

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

   3. pow2 99.7%

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

   4. pow-flip 99.8%

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

   5. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.4 \lor \neg \left(v \leq 3.6 \cdot 10^{-51}\right):\\ \;\;\;\;t\_0 - \left(4.5 - \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r r)))))
   (if (or (<= v -1.4) (not (<= v 3.6e-51)))
     (- t_0 (- 4.5 (* (* v -0.25) (* (* r w) (/ (* r w) v)))))
     (- t_0 (+ 4.5 (* 0.375 (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -1.4) || !(v <= 3.6e-51)) {
		tmp = t_0 - (4.5 - ((v * -0.25) * ((r * w) * ((r * w) / v))));
	} else {
		tmp = t_0 - (4.5 + (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 + (2.0d0 / (r * r))
    if ((v <= (-1.4d0)) .or. (.not. (v <= 3.6d-51))) then
        tmp = t_0 - (4.5d0 - ((v * (-0.25d0)) * ((r * w) * ((r * w) / v))))
    else
        tmp = t_0 - (4.5d0 + (0.375d0 * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 3.0 + (2.0 / (r * r));
	double tmp;
	if ((v <= -1.4) || !(v <= 3.6e-51)) {
		tmp = t_0 - (4.5 - ((v * -0.25) * ((r * w) * ((r * w) / v))));
	} else {
		tmp = t_0 - (4.5 + (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 3.0 + (2.0 / (r * r))
	tmp = 0
	if (v <= -1.4) or not (v <= 3.6e-51):
		tmp = t_0 - (4.5 - ((v * -0.25) * ((r * w) * ((r * w) / v))))
	else:
		tmp = t_0 - (4.5 + (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if ((v <= -1.4) || !(v <= 3.6e-51))
		tmp = Float64(t_0 - Float64(4.5 - Float64(Float64(v * -0.25) * Float64(Float64(r * w) * Float64(Float64(r * w) / v)))));
	else
		tmp = Float64(t_0 - Float64(4.5 + Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 3.0 + (2.0 / (r * r));
	tmp = 0.0;
	if ((v <= -1.4) || ~((v <= 3.6e-51)))
		tmp = t_0 - (4.5 - ((v * -0.25) * ((r * w) * ((r * w) / v))));
	else
		tmp = t_0 - (4.5 + (0.375 * ((r * w) * (r * w))));
	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[Or[LessEqual[v, -1.4], N[Not[LessEqual[v, 3.6e-51]], $MachinePrecision]], N[(t$95$0 - N[(4.5 - N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(4.5 + N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1.3999999999999999 or 3.6e-51 < v

   1. Initial program 83.6%

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

   3. Simplified 88.3%

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

      1. associate-/l* 87.7%

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

      2. *-commutative 87.7%

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

      3. associate-*r/ 86.7%

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

      4. associate-*l* 94.9%

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

      5. associate-*r* 98.7%

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

      6. *-commutative 98.7%

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

      7. associate-*r/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around inf 96.9%

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

      1. mul-1-neg 96.9%

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

   9. Simplified 96.9%

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

   10. Taylor expanded in v around inf 99.0%

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

   if -1.3999999999999999 < v < 3.6e-51

   1. Initial program 84.2%

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

   3. Simplified 84.2%

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

      1. associate-/l* 84.2%

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

      2. *-commutative 84.2%

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

      3. associate-*r/ 84.2%

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

      4. associate-*l* 94.7%

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

      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. *-commutative 99.8%

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

      7. associate-*r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in v around 0 99.8%

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

   8. Taylor expanded in v around 0 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.4 \lor \neg \left(v \leq 3.6 \cdot 10^{-51}\right):\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 - \left(v \cdot -0.25\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r \cdot w}{v}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \left(4.5 + 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (/ (+ (* v -0.25) 0.375) (/ (+ v -1.0) (* (* r w) (* r w)))))))

C

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

Fortran

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

Java

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / ((v + -1.0) / ((r * w) * (r * w)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.6%

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

   1. fma-undefine 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   2. *-commutative 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   3. +-commutative 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   4. associate-*r/ 86.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

   5. *-commutative 86.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

   6. associate-/l* 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

   7. clear-num 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   8. un-div-inv 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   9. +-commutative 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. distribute-rgt-in 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 86.5%

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

   15. associate-*r* 81.0%

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

   16. pow2 81.0%

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

   17. pow2 81.0%

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

   18. pow-prod-down 99.7%

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

   19. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (/ (+ (* v -0.25) 0.375) (/ (/ (+ v -1.0) (* r w)) (* r w))))))

C

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

Fortran

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

Java

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (((v * -0.25) + 0.375) / (((v + -1.0) / (r * w)) / (r * w))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

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

3. Simplified 85.6%

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

   1. fma-undefine 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(v \cdot -2 + 3\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   2. *-commutative 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(\color{blue}{-2 \cdot v} + 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   3. +-commutative 85.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \color{blue}{\left(3 + -2 \cdot v\right)}\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right) \]

   4. associate-*r/ 86.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \color{blue}{\frac{\left(w \cdot w\right) \cdot r}{1 - v}}\right)\right) \]

   5. *-commutative 86.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(r \cdot \frac{\color{blue}{r \cdot \left(w \cdot w\right)}}{1 - v}\right)\right) \]

   6. associate-/l* 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}{1 - v}}\right) \]

   7. clear-num 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \color{blue}{\frac{1}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   8. un-div-inv 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}}\right) \]

   9. +-commutative 86.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.125 \cdot \color{blue}{\left(-2 \cdot v + 3\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   10. distribute-rgt-in 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(-2 \cdot v\right) \cdot 0.125 + 3 \cdot 0.125}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(v \cdot -2\right)} \cdot 0.125 + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{v \cdot \left(-2 \cdot 0.125\right)} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 86.5%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{v \cdot \color{blue}{-0.25} + 3 \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   14. metadata-eval 86.5%

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

   15. associate-*r* 81.0%

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

   16. pow2 81.0%

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

   17. pow2 81.0%

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

   18. pow-prod-down 99.7%

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

   19. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-/r* 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. times-frac 95.3%

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

10. Applied egg-rr 95.3%

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

    1. frac-times 99.7%

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

    2. *-un-lft-identity 99.7%

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

    3. *-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 5: 93.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + (2.0 / (r * r))) - (4.5 + (0.375 * ((r * w) * (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[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.9%

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

3. Simplified 86.5%

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

   1. associate-/l* 86.1%

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

   2. *-commutative 86.1%

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

   3. associate-*r/ 85.6%

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

   4. associate-*l* 94.8%

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

   5. associate-*r* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r/ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0 80.7%

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

8. Taylor expanded in v around 0 92.5%

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

9. Final simplification 92.5%

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

Reproduce

herbie shell --seed 2024059 
(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))