Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.7%

Time: 11.6s

Alternatives: 10

Speedup: 1.7×

• 53.2% 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: 84.8% 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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

Java

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

Python

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

Julia

function code(v, w, r)
	return Float64(Float64(3.0 + Float64(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(v + -1.0)))) - 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) / (v + -1.0)))) - 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[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

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

   1. associate--l- 80.0%

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

   2. associate-*l* 75.1%

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

   3. sqr-neg 75.1%

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

   4. associate-*l* 80.0%

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

   5. associate-/l* 84.0%

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

   6. fma-define 84.0%

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

3. Simplified 84.0%

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

   1. add-sqr-sqrt 84.0%

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

   2. *-un-lft-identity 84.0%

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

   3. times-frac 84.0%

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

   4. associate-*r* 77.7%

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

   5. sqrt-prod 77.7%

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

   6. sqrt-prod 40.2%

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

   7. add-sqr-sqrt 69.8%

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

   8. sqrt-prod 29.0%

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

   9. add-sqr-sqrt 66.0%

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

   10. associate-*r* 58.6%

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

   11. sqrt-prod 58.6%

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

   12. sqrt-prod 30.9%

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

   13. add-sqr-sqrt 69.1%

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

   14. sqrt-prod 44.4%

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

   15. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. associate--l- 80.0%

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

   2. associate-*l* 75.1%

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

   3. sqr-neg 75.1%

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

   4. associate-*l* 80.0%

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

   5. associate-/l* 84.0%

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

   6. fma-define 84.0%

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

3. Simplified 84.0%

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

   1. associate-/l* 84.0%

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

   2. *-commutative 84.0%

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

   3. associate-*r/ 83.9%

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

   4. associate-*l* 95.3%

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

   5. associate-*r* 99.0%

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

6. Applied egg-rr 99.0%

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

   1. associate-/r* 99.1%

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

   2. div-inv 99.0%

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

8. Applied egg-rr 99.0%

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

   1. associate-*r/ 99.1%

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

   2. *-rgt-identity 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. associate--l- 80.0%

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

   2. associate-*l* 75.1%

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

   3. sqr-neg 75.1%

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

   4. associate-*l* 80.0%

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

   5. associate-/l* 84.0%

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

   6. fma-define 84.0%

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

3. Simplified 84.0%

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

   1. associate-/l* 84.0%

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

   2. *-commutative 84.0%

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

   3. associate-*r/ 83.9%

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

   4. associate-*l* 95.3%

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

   5. associate-*r* 99.0%

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

6. Applied egg-rr 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00044:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.00044)
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* (* r w) (* r w)) 0.25)))
   (-
    3.0
    (+
     4.5
     (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w (/ r (- 1.0 v)))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00044) {
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))));
	}
	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 <= 0.00044d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * (r / (1.0d0 - v))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00044) {
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00044:
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25))
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00044)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(w * Float64(r / Float64(1.0 - v)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00044)
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * (r / (1.0 - v))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00044], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.40000000000000016e-4

   1. Initial program 78.5%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in v around inf 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. unpow2 73.5%

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

      4. swap-sqr 90.7%

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

      5. unpow2 90.7%

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

   7. Simplified 90.7%

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

      1. unpow2 90.7%

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

   9. Applied egg-rr 90.7%

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

   if 4.40000000000000016e-4 < r

   1. Initial program 86.1%

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

      1. associate--l- 86.1%

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

      2. associate-*l* 74.1%

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

      3. sqr-neg 74.1%

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

      4. associate-*l* 86.1%

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

      5. associate-/l* 91.8%

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

      6. fma-define 91.9%

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

   3. Simplified 91.8%

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

      1. associate-/l* 91.9%

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

      2. *-commutative 91.9%

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

      3. associate-*r/ 91.9%

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

      4. associate-*l* 95.7%

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

      5. associate-*r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.00044:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{1 - v}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

   1. associate--l- 80.0%

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

   2. associate-*l* 75.1%

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

   3. sqr-neg 75.1%

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

   4. associate-*l* 80.0%

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

   5. associate-/l* 84.0%

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

   6. fma-define 84.0%

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

3. Simplified 84.0%

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

   1. add-sqr-sqrt 84.0%

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

   2. *-un-lft-identity 84.0%

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

   3. times-frac 84.0%

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

   4. associate-*r* 77.7%

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

   5. sqrt-prod 77.7%

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

   6. sqrt-prod 40.2%

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

   7. add-sqr-sqrt 69.8%

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

   8. sqrt-prod 29.0%

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

   9. add-sqr-sqrt 66.0%

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

   10. associate-*r* 58.6%

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

   11. sqrt-prod 58.6%

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

   12. sqrt-prod 30.9%

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

   13. add-sqr-sqrt 69.1%

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

   14. sqrt-prod 44.4%

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

   15. add-sqr-sqrt 99.4%

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

6. Applied egg-rr 99.4%

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

   1. /-rgt-identity 99.4%

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

   2. associate-*r* 97.6%

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

   3. clear-num 97.5%

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

   4. un-div-inv 97.5%

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

   5. *-commutative 97.5%

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

   6. distribute-rgt-in 97.5%

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

   7. metadata-eval 97.5%

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

   8. *-commutative 97.5%

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

   9. associate-*l* 97.5%

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

   10. metadata-eval 97.5%

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

8. Applied egg-rr 97.5%

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

9. Final simplification 97.5%

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

Alternative 6: 95.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1800000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1800000000.0)
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* (* r w) (* r w)) 0.25)))
   (-
    (+ 3.0 (* (* r (* w (+ 0.375 (* v -0.25)))) (* w (/ r (+ v -1.0)))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1800000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1800000000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
    else
        tmp = (3.0d0 + ((r * (w * (0.375d0 + (v * (-0.25d0))))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1800000000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1800000000.0:
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25))
	else:
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1800000000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25)))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1800000000.0)
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	else
		tmp = (3.0 + ((r * (w * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1800000000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.8e9

   1. Initial program 78.3%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in v around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. unpow2 73.4%

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

      3. unpow2 73.4%

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

      4. swap-sqr 90.4%

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

      5. unpow2 90.4%

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

   7. Simplified 90.4%

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

      1. unpow2 90.4%

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

   9. Applied egg-rr 90.4%

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

   if 1.8e9 < r

   1. Initial program 87.6%

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

   3. Taylor expanded in r around inf 87.6%

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

      1. associate-/l* 91.6%

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

      2. cancel-sign-sub-inv 91.6%

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

      3. metadata-eval 91.6%

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

      4. +-commutative 91.6%

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

      5. *-commutative 91.6%

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

      6. fma-undefine 91.6%

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

      7. *-commutative 91.6%

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

      8. *-commutative 91.6%

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

      9. associate-/l* 91.6%

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

      10. *-commutative 91.6%

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

      11. associate-*r/ 91.6%

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

      12. associate-*r* 85.4%

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

      13. associate-*l* 89.3%

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

      14. associate-*r* 89.5%

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

   5. Applied egg-rr 89.5%

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

   6. Taylor expanded in r around 0 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1800000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 16500000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{v + -1}{r \cdot w}} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 16500000.0)
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* (* r w) (* r w)) 0.25)))
   (+
    3.0
    (- (/ (* (* r w) (+ 0.375 (* v -0.25))) (/ (+ v -1.0) (* r w))) 4.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 16500000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * 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 <= 16500000.0d0) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
    else
        tmp = 3.0d0 + ((((r * w) * (0.375d0 + (v * (-0.25d0)))) / ((v + (-1.0d0)) / (r * w))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 16500000.0) {
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 16500000.0:
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25))
	else:
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 16500000.0)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(Float64(r * w) * Float64(0.375 + Float64(v * -0.25))) / Float64(Float64(v + -1.0) / Float64(r * w))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 16500000.0)
		tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) * (r * w)) * 0.25));
	else
		tmp = 3.0 + ((((r * w) * (0.375 + (v * -0.25))) / ((v + -1.0) / (r * w))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 16500000.0], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(N[(r * w), $MachinePrecision] * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.65e7

   1. Initial program 78.3%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in v around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. unpow2 73.4%

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

      3. unpow2 73.4%

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

      4. swap-sqr 90.4%

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

      5. unpow2 90.4%

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

   7. Simplified 90.4%

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

      1. unpow2 90.4%

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

   9. Applied egg-rr 90.4%

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

   if 1.65e7 < r

   1. Initial program 87.6%

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

      1. associate--l- 87.6%

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

      2. associate-*l* 75.1%

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

      3. sqr-neg 75.1%

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

      4. associate-*l* 87.6%

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

      5. associate-/l* 91.6%

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

      6. fma-define 91.6%

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

   3. Simplified 91.6%

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

      1. add-sqr-sqrt 91.5%

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

      2. *-un-lft-identity 91.5%

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

      3. times-frac 91.5%

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

      4. associate-*r* 79.0%

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

      5. sqrt-prod 79.0%

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

      6. sqrt-prod 91.5%

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

      7. add-sqr-sqrt 91.6%

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

      8. sqrt-prod 38.1%

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

      9. add-sqr-sqrt 47.0%

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

      10. associate-*r* 34.5%

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

      11. sqrt-prod 34.5%

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

      12. sqrt-prod 47.0%

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

      13. add-sqr-sqrt 47.0%

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

      14. sqrt-prod 42.5%

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

      15. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. /-rgt-identity 99.8%

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

      2. associate-*r* 99.8%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-in 99.8%

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

      7. metadata-eval 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*l* 99.8%

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

      10. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 16500000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{\frac{v + -1}{r \cdot w}} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 83.9%

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

5. Taylor expanded in v around inf 73.8%

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

   1. *-commutative 73.8%

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

   2. unpow2 73.8%

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

   3. unpow2 73.8%

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

   4. swap-sqr 90.2%

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

   5. unpow2 90.2%

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

7. Simplified 90.2%

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

   1. unpow2 90.2%

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

9. Applied egg-rr 90.2%

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

Alternative 9: 57.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{r}}{r} + -1.5 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.0%

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

3. Simplified 77.3%

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

5. Taylor expanded in r around 0 59.4%

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

   1. +-commutative 59.4%

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

   2. associate--l+ 59.4%

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

   3. associate-/r* 59.4%

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

   4. div-inv 59.3%

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

   5. div-inv 59.3%

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

   6. unpow-1 59.3%

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

   7. metadata-eval 59.3%

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

   8. unpow-1 59.3%

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

   9. metadata-eval 59.3%

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

   10. associate-*r* 59.3%

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

   11. sqr-pow 59.4%

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

   12. metadata-eval 59.4%

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

7. Applied egg-rr 59.4%

   \[\leadsto \color{blue}{2 \cdot {r}^{-2} + -1.5} \]
8. Step-by-step derivation

   1. sqr-pow 59.3%

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

   2. associate-*r* 59.3%

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

   3. metadata-eval 59.3%

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

   4. unpow-1 59.3%

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

   5. div-inv 59.3%

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

   6. metadata-eval 59.3%

      \[\leadsto \frac{2}{r} \cdot {r}^{\color{blue}{-1}} + -1.5 \]

   7. unpow-1 59.3%

      \[\leadsto \frac{2}{r} \cdot \color{blue}{\frac{1}{r}} + -1.5 \]

   8. div-inv 59.4%

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

9. Applied egg-rr 59.4%

   \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + -1.5 \]
10. Add Preprocessing

Alternative 10: 14.0% 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 80.0%

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

3. Simplified 77.3%

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

5. Taylor expanded in r around 0 59.4%

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

6. Taylor expanded in r around inf 11.0%

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

Reproduce

herbie shell --seed 2024136 
(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))