Rosa's TurbineBenchmark

Percentage Accurate: 85.4% → 99.8%

Time: 14.7s

Alternatives: 8

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 1.0× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r r)))
  (+ (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (/ (* r w) (- 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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (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))) * ((r * w) * ((r * w) / (1.0 - v)))) + 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(Float64(r * w) / 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))) * ((r * w) * ((r * w) / (1.0 - v)))) + 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[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. associate--l- 83.1%

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

   2. associate-*l* 80.1%

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

   3. sqr-neg 80.1%

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

   4. associate-*l* 83.1%

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

   5. associate-/l* 86.8%

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

   6. fma-define 86.8%

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

3. Simplified 86.8%

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

   1. add-sqr-sqrt 86.7%

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

   2. *-un-lft-identity 86.7%

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

   3. times-frac 86.8%

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

   4. *-commutative 86.8%

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

   5. sqrt-prod 44.7%

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

   6. *-commutative 44.7%

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

   7. sqrt-prod 44.7%

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

   8. sqrt-prod 19.0%

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

   9. add-sqr-sqrt 36.0%

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

   10. associate-*r* 36.0%

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

   11. add-sqr-sqrt 69.3%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 85.7%

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

   1. fma-undefine 85.7%

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

   2. *-commutative 85.7%

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

   3. +-commutative 85.7%

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

   4. metadata-eval 85.7%

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

   5. cancel-sign-sub-inv 85.7%

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

   6. associate-*r/ 86.1%

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

   7. *-commutative 86.1%

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

   8. associate-/l* 86.8%

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

   9. clear-num 86.8%

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

   10. un-div-inv 86.8%

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

   11. cancel-sign-sub-inv 86.8%

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

   12. metadata-eval 86.8%

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

   13. distribute-rgt-in 86.8%

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

   14. metadata-eval 86.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. *-commutative 86.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   16. associate-*l* 86.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   17. metadata-eval 86.8%

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

   18. *-commutative 86.8%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

   2. /-rgt-identity 99.7%

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

   3. /-rgt-identity 99.7%

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

   4. clear-num 99.7%

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

   5. frac-times 99.7%

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

   6. metadata-eval 99.7%

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

   7. div-inv 99.7%

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

   8. /-rgt-identity 99.7%

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

8. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. times-frac 95.7%

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

   3. associate-/r* 95.6%

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

   4. /-rgt-identity 95.6%

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

10. Simplified 95.6%

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

    1. associate-*l/ 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ -1.5 (/ (+ 0.375 (* v -0.25)) (/ (+ 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 + (v * -0.25)) / ((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 + (v * (-0.25d0))) / ((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 + (v * -0.25)) / ((v + -1.0) / ((r * w) * (r * w)))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((0.375 + (v * -0.25)) / ((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(v * -0.25)) / 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 + (v * -0.25)) / ((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[(v * -0.25), $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 83.1%

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

3. Simplified 85.7%

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

   1. fma-undefine 85.7%

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

   2. *-commutative 85.7%

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

   3. +-commutative 85.7%

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

   4. metadata-eval 85.7%

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

   5. cancel-sign-sub-inv 85.7%

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

   6. associate-*r/ 86.1%

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

   7. *-commutative 86.1%

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

   8. associate-/l* 86.8%

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

   9. clear-num 86.8%

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

   10. un-div-inv 86.8%

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

   11. cancel-sign-sub-inv 86.8%

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

   12. metadata-eval 86.8%

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

   13. distribute-rgt-in 86.8%

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

   14. metadata-eval 86.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   15. *-commutative 86.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   16. associate-*l* 86.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   17. metadata-eval 86.8%

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

   18. *-commutative 86.8%

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

6. Applied egg-rr 99.7%

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

   1. unpow2 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 82.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) + (((r * w) * 0.375) * (w / ((v + -1.0) / r)))) - 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[(r * w), $MachinePrecision] * 0.375), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 83.1%

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

   1. associate-/l* 86.8%

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

   2. cancel-sign-sub-inv 86.8%

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

   3. metadata-eval 86.8%

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

   4. +-commutative 86.8%

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

   5. *-commutative 86.8%

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

   6. fma-undefine 86.8%

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

   7. *-commutative 86.8%

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

   8. *-commutative 86.8%

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

   9. associate-/l* 86.1%

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

   10. *-commutative 86.1%

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

   11. associate-*r/ 85.7%

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

   12. associate-*r* 83.0%

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

   13. associate-*l* 93.0%

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

   14. associate-*r* 93.8%

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

4. Applied egg-rr 93.9%

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

5. Taylor expanded in v around 0 81.2%

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

6. Final simplification 81.2%

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

Alternative 5: 82.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. associate-/l* 86.8%

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

   2. cancel-sign-sub-inv 86.8%

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

   3. metadata-eval 86.8%

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

   4. +-commutative 86.8%

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

   5. *-commutative 86.8%

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

   6. fma-undefine 86.8%

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

   7. *-commutative 86.8%

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

   8. *-commutative 86.8%

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

   9. associate-/l* 86.1%

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

   10. *-commutative 86.1%

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

   11. associate-*r/ 85.7%

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

   12. associate-*r* 83.0%

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

   13. associate-*l* 93.0%

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

   14. associate-*r* 93.8%

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

4. Applied egg-rr 93.9%

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

5. Taylor expanded in v around 0 81.2%

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

   1. associate-*r* 81.2%

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

7. Simplified 81.2%

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

   1. associate-/r/ 81.2%

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

9. Applied egg-rr 81.2%

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

10. Final simplification 81.2%

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

Alternative 6: 57.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 81.4%

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

5. Taylor expanded in r around 0 56.6%

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

6. Final simplification 56.6%

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

Alternative 7: 57.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 81.4%

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

5. Taylor expanded in r around 0 56.6%

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

   1. associate-/r* 56.6%

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

   2. div-inv 56.6%

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

7. Applied egg-rr 56.6%

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

   1. associate-*r/ 56.6%

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

   2. *-rgt-identity 56.6%

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

9. Simplified 56.6%

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

10. Final simplification 56.6%

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

Alternative 8: 14.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

3. Simplified 81.4%

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

5. Taylor expanded in r around 0 56.6%

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

6. Taylor expanded in r around inf 10.2%

   \[\leadsto \color{blue}{-1.5} \]

7. Final simplification 10.2%

   \[\leadsto -1.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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))