Rosa's TurbineBenchmark

Percentage Accurate: 84.1% → 99.8%

Time: 11.9s

Alternatives: 6

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: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((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) + ((0.375d0 + ((-0.25d0) * v)) / ((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 + ((0.375 + (-0.25 * v)) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (-0.25 * v)) / ((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(0.375 + Float64(-0.25 * v)) / 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 + ((0.375 + (-0.25 * v)) / ((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[(0.375 + N[(-0.25 * v), $MachinePrecision]), $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 84.0%

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

3. Simplified 86.2%

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

   1. associate-*l* 86.2%

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

   2. fma-undefine 86.2%

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

   3. *-commutative 86.2%

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

   4. +-commutative 86.2%

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

   5. metadata-eval 86.2%

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

   6. cancel-sign-sub-inv 86.2%

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

   7. associate-*r/ 86.6%

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

   8. *-commutative 86.6%

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

   9. associate-/l* 86.6%

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

   10. associate-*l* 86.6%

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

   11. clear-num 86.6%

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

   12. un-div-inv 86.7%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1 \cdot 10^{+34} \lor \neg \left(v \leq 1\right):\\ \;\;\;\;t\_0 + \left(-1.5 + \frac{0.375 + -0.25 \cdot v}{\frac{v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -1e+34) (not (<= v 1.0)))
     (+ t_0 (+ -1.5 (/ (+ 0.375 (* -0.25 v)) (/ v (* (* r w) (* r w))))))
     (- (- (+ t_0 3.0) (* (* r w) (* r (* 0.375 w)))) 4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1e+34) || !(v <= 1.0)) {
		tmp = t_0 + (-1.5 + ((0.375 + (-0.25 * v)) / (v / ((r * w) * (r * w)))));
	} else {
		tmp = ((t_0 + 3.0) - ((r * w) * (r * (0.375 * w)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-1d+34)) .or. (.not. (v <= 1.0d0))) then
        tmp = t_0 + ((-1.5d0) + ((0.375d0 + ((-0.25d0) * v)) / (v / ((r * w) * (r * w)))))
    else
        tmp = ((t_0 + 3.0d0) - ((r * w) * (r * (0.375d0 * w)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -1e+34) || !(v <= 1.0)) {
		tmp = t_0 + (-1.5 + ((0.375 + (-0.25 * v)) / (v / ((r * w) * (r * w)))));
	} else {
		tmp = ((t_0 + 3.0) - ((r * w) * (r * (0.375 * w)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -1e+34) or not (v <= 1.0):
		tmp = t_0 + (-1.5 + ((0.375 + (-0.25 * v)) / (v / ((r * w) * (r * w)))))
	else:
		tmp = ((t_0 + 3.0) - ((r * w) * (r * (0.375 * w)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -1e+34) || !(v <= 1.0))
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(0.375 + Float64(-0.25 * v)) / Float64(v / Float64(Float64(r * w) * Float64(r * w))))));
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) - Float64(Float64(r * w) * Float64(r * Float64(0.375 * w)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -1e+34) || ~((v <= 1.0)))
		tmp = t_0 + (-1.5 + ((0.375 + (-0.25 * v)) / (v / ((r * w) * (r * w)))));
	else
		tmp = ((t_0 + 3.0) - ((r * w) * (r * (0.375 * w)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -1e+34], N[Not[LessEqual[v, 1.0]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 + N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] / N[(v / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(r * N[(0.375 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -9.99999999999999946e33 or 1 < v

   1. Initial program 77.1%

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

   3. Simplified 81.9%

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

      1. associate-*l* 81.9%

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

      2. fma-undefine 81.9%

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

      3. *-commutative 81.9%

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

      4. +-commutative 81.9%

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

      5. metadata-eval 81.9%

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

      6. cancel-sign-sub-inv 81.9%

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

      7. associate-*r/ 82.7%

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

      8. *-commutative 82.7%

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

      9. associate-/l* 82.7%

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

      10. associate-*l* 82.7%

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

      11. clear-num 82.7%

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

      12. un-div-inv 82.7%

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

   6. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in v around inf 99.2%

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

       1. neg-mul-1 99.2%

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

   11. Simplified 99.2%

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

   if -9.99999999999999946e33 < v < 1

   1. Initial program 90.0%

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

      1. associate-/l* 90.0%

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

      2. cancel-sign-sub-inv 90.0%

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

      3. metadata-eval 90.0%

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

      4. +-commutative 90.0%

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

      5. *-commutative 90.0%

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

      6. fma-undefine 90.0%

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

      7. *-commutative 90.0%

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

      8. *-commutative 90.0%

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

      9. associate-/l* 90.0%

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

      10. *-commutative 90.0%

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

      11. associate-*r/ 90.0%

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

      12. associate-*r* 90.0%

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

      13. associate-*l* 98.4%

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

      14. associate-*r* 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-*l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in v around 0 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1 \cdot 10^{+34} \lor \neg \left(v \leq 1\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + -0.25 \cdot v}{\frac{v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(r \cdot w\right) \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

   1. associate-/l* 86.6%

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

   2. cancel-sign-sub-inv 86.6%

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

   3. metadata-eval 86.6%

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

   4. +-commutative 86.6%

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

   5. *-commutative 86.6%

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

   6. fma-undefine 86.6%

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

   7. *-commutative 86.6%

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

   8. *-commutative 86.6%

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

   9. associate-/l* 86.6%

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

   10. *-commutative 86.6%

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

   11. associate-*r/ 86.3%

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

   12. associate-*r* 83.8%

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

   13. associate-*l* 93.8%

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

   14. associate-*r* 93.1%

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

4. Applied egg-rr 93.1%

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

5. Taylor expanded in v around 0 84.8%

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

   1. *-commutative 84.8%

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

   2. *-commutative 84.8%

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

   3. *-commutative 84.8%

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

   4. associate-*l* 84.9%

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

7. Simplified 84.9%

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

8. Taylor expanded in v around 0 94.1%

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

9. Final simplification 94.1%

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

Alternative 4: 72.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9.5:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 9.5)
   (- (+ (/ 2.0 (* r r)) 3.0) 4.5)
   (- (- 3.0 (* (* r w) (* r (* 0.375 w)))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.5) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else {
		tmp = (3.0 - ((r * w) * (r * (0.375 * w)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 9.5d0) then
        tmp = ((2.0d0 / (r * r)) + 3.0d0) - 4.5d0
    else
        tmp = (3.0d0 - ((r * w) * (r * (0.375d0 * w)))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.5) {
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	} else {
		tmp = (3.0 - ((r * w) * (r * (0.375 * w)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 9.5:
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5
	else:
		tmp = (3.0 - ((r * w) * (r * (0.375 * w)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9.5)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + 3.0) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r * w) * Float64(r * Float64(0.375 * w)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 9.5)
		tmp = ((2.0 / (r * r)) + 3.0) - 4.5;
	else
		tmp = (3.0 - ((r * w) * (r * (0.375 * w)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9.5], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r * w), $MachinePrecision] * N[(r * N[(0.375 * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.5

   1. Initial program 81.3%

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in r around 0 68.6%

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

   if 9.5 < r

   1. Initial program 94.5%

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

      1. associate-/l* 94.5%

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

      2. cancel-sign-sub-inv 94.5%

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

      3. metadata-eval 94.5%

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

      4. +-commutative 94.5%

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

      5. *-commutative 94.5%

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

      6. fma-undefine 94.5%

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

      7. *-commutative 94.5%

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

      8. *-commutative 94.5%

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

      9. associate-/l* 94.4%

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

      10. *-commutative 94.4%

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

      11. associate-*r/ 94.5%

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

      12. associate-*r* 90.5%

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

      13. associate-*l* 90.8%

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

      14. associate-*r* 90.7%

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

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in r around inf 90.7%

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

   6. Taylor expanded in v around 0 79.1%

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

   7. Taylor expanded in v around 0 91.4%

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

      1. *-commutative 78.9%

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

      2. *-commutative 78.9%

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

      3. *-commutative 78.9%

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

      4. associate-*l* 79.0%

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

   9. Simplified 91.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9.5:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(0.375 \cdot w\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

3. Simplified 79.2%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 61.2%

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

6. Final simplification 61.2%

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

Alternative 6: 13.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.0%

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

3. Simplified 79.2%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 61.2%

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

6. Taylor expanded in r around inf 14.9%

   \[\leadsto \color{blue}{-1.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024116 
(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))