Rosa's TurbineBenchmark

Percentage Accurate: 84.7% → 99.8%

Time: 16.4s

Alternatives: 6

Speedup: 1.7×

• 52.9% 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: 84.7% 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, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (3.0 + (2.0 / (r * r))) + (((0.375 + (-0.25 * v)) / (((v + -1.0) / (r * w)) / (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[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(v + -1.0), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. associate--l- 85.5%

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

   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* 85.5%

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

   5. associate-/l* 88.4%

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

   6. fma-define 88.4%

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

3. Simplified 88.4%

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

   1. associate-/l* 88.4%

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

   2. *-commutative 88.4%

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

   3. associate-*r/ 87.6%

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

   4. associate-*l* 97.2%

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

   5. associate-*r* 98.7%

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

   6. add-sqr-sqrt 47.9%

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

   7. associate-*l* 47.9%

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

   8. add-sqr-sqrt 24.6%

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

   9. sqrt-prod 34.4%

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

   10. sqrt-prod 34.4%

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

   11. sqrt-prod 68.4%

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

   12. *-commutative 68.4%

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

   13. sqrt-prod 34.4%

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

   14. *-commutative 34.4%

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

   15. sqrt-prod 34.4%

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

   16. sqrt-prod 24.6%

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

   17. add-sqr-sqrt 47.9%

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

   18. associate-*r* 47.9%

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

   19. add-sqr-sqrt 98.7%

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

   20. clear-num 98.6%

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

   21. un-div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

   1. clear-num 98.7%

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

   2. un-div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

   1. clear-num 98.7%

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

   2. un-div-inv 98.7%

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

   3. distribute-lft-in 98.7%

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

   4. metadata-eval 98.7%

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

   5. associate-*r* 98.7%

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

   6. metadata-eval 98.7%

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

   7. associate-/l/ 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -5 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;\left(3 + t\_0\right) - \left(4.5 - \frac{0.375 + -0.25 \cdot v}{\frac{\frac{v}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -5.0) (not (<= v 1.0)))
     (-
      (+ 3.0 t_0)
      (- 4.5 (/ (+ 0.375 (* -0.25 v)) (/ (/ v (* r w)) (* r w)))))
     (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -5.0) || !(v <= 1.0)) {
		tmp = (3.0 + t_0) - (4.5 - ((0.375 + (-0.25 * v)) / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((v <= (-5.0d0)) .or. (.not. (v <= 1.0d0))) then
        tmp = (3.0d0 + t_0) - (4.5d0 - ((0.375d0 + ((-0.25d0) * v)) / ((v / (r * w)) / (r * w))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -5.0) || !(v <= 1.0)) {
		tmp = (3.0 + t_0) - (4.5 - ((0.375 + (-0.25 * v)) / ((v / (r * w)) / (r * w))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -5.0) or not (v <= 1.0):
		tmp = (3.0 + t_0) - (4.5 - ((0.375 + (-0.25 * v)) / ((v / (r * w)) / (r * w))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -5.0) || !(v <= 1.0))
		tmp = Float64(Float64(3.0 + t_0) - Float64(4.5 - Float64(Float64(0.375 + Float64(-0.25 * v)) / Float64(Float64(v / Float64(r * w)) / Float64(r * w)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -5.0) || ~((v <= 1.0)))
		tmp = (3.0 + t_0) - (4.5 - ((0.375 + (-0.25 * v)) / ((v / (r * w)) / (r * w))));
	else
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -5.0], N[Not[LessEqual[v, 1.0]], $MachinePrecision]], N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(4.5 - N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] / N[(N[(v / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -5 or 1 < v

   1. Initial program 83.0%

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

      1. associate--l- 83.0%

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

      2. associate-*l* 78.0%

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

      3. sqr-neg 78.0%

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

      4. associate-*l* 83.0%

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

      5. associate-/l* 88.1%

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

      6. fma-define 88.1%

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

   3. Simplified 88.1%

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

      1. associate-/l* 88.1%

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

      2. *-commutative 88.1%

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

      3. associate-*r/ 86.7%

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

      4. associate-*l* 95.8%

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

      5. associate-*r* 97.7%

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

      6. add-sqr-sqrt 43.5%

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

      7. associate-*l* 43.6%

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

      8. add-sqr-sqrt 18.3%

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

      9. sqrt-prod 31.0%

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

      10. sqrt-prod 31.0%

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

      11. sqrt-prod 65.1%

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

      12. *-commutative 65.1%

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

      13. sqrt-prod 31.0%

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

      14. *-commutative 31.0%

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

      15. sqrt-prod 31.0%

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

      16. sqrt-prod 18.3%

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

      17. add-sqr-sqrt 43.6%

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

      18. associate-*r* 43.5%

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

      19. add-sqr-sqrt 97.7%

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

      20. clear-num 97.7%

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

      21. un-div-inv 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. clear-num 97.7%

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

      2. un-div-inv 97.7%

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

   8. Applied egg-rr 97.7%

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

      1. clear-num 97.7%

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

      2. un-div-inv 97.7%

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

      3. distribute-lft-in 97.7%

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

      4. metadata-eval 97.7%

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

      5. associate-*r* 97.7%

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

      6. metadata-eval 97.7%

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

      7. associate-/l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in v around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. *-commutative 98.7%

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

       3. distribute-neg-frac2 98.7%

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

       4. *-commutative 98.7%

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

       5. distribute-rgt-neg-in 98.7%

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

   13. Simplified 98.7%

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

   if -5 < v < 1

   1. Initial program 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in v around 0 82.1%

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

      1. *-commutative 82.1%

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

      2. unpow2 82.1%

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

      3. unpow2 82.1%

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

      4. swap-sqr 99.3%

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

      5. unpow2 99.3%

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

      6. *-commutative 99.3%

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

   7. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. pow2 99.3%

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

   9. Applied egg-rr 99.3%

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

4. Final simplification 99.0%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. associate--l- 85.5%

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

   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* 85.5%

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

   5. associate-/l* 88.4%

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

   6. fma-define 88.4%

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

3. Simplified 88.4%

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

   1. associate-/l* 88.4%

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

   2. *-commutative 88.4%

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

   3. associate-*r/ 87.6%

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

   4. associate-*l* 97.2%

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

   5. associate-*r* 98.7%

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

   6. add-sqr-sqrt 47.9%

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

   7. associate-*l* 47.9%

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

   8. add-sqr-sqrt 24.6%

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

   9. sqrt-prod 34.4%

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

   10. sqrt-prod 34.4%

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

   11. sqrt-prod 68.4%

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

   12. *-commutative 68.4%

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

   13. sqrt-prod 34.4%

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

   14. *-commutative 34.4%

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

   15. sqrt-prod 34.4%

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

   16. sqrt-prod 24.6%

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

   17. add-sqr-sqrt 47.9%

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

   18. associate-*r* 47.9%

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

   19. add-sqr-sqrt 98.7%

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

   20. clear-num 98.6%

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

   21. un-div-inv 98.7%

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

6. Applied egg-rr 98.7%

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

   1. clear-num 98.7%

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

   2. un-div-inv 98.7%

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

8. Applied egg-rr 98.7%

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

   1. *-commutative 98.7%

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

   2. distribute-lft-in 98.7%

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

   3. metadata-eval 98.7%

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

   4. distribute-lft-in 86.9%

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

   5. associate-/r/ 85.1%

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

   6. associate-/r/ 85.1%

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

   7. associate-*r* 85.1%

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

   8. metadata-eval 85.1%

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

10. Applied egg-rr 85.1%

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

    1. distribute-lft-out 97.2%

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

    2. metadata-eval 97.2%

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

    3. associate-*r* 97.2%

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

    4. *-commutative 97.2%

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

    5. *-commutative 97.2%

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

    6. associate-/l* 92.4%

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

    7. *-commutative 92.4%

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

    8. associate-*l* 92.4%

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

    9. metadata-eval 92.4%

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

12. Simplified 92.4%

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

    1. associate-*r/ 97.2%

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

14. Applied egg-rr 97.2%

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

15. Final simplification 97.2%

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

Alternative 4: 93.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Simplified 87.6%

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

5. Taylor expanded in v around 0 77.4%

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

   1. *-commutative 77.4%

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

   2. unpow2 77.4%

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

   3. unpow2 77.4%

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

   4. swap-sqr 91.4%

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

   5. unpow2 91.4%

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

   6. *-commutative 91.4%

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

7. Simplified 91.4%

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

   1. *-commutative 91.4%

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

   2. pow2 91.4%

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

9. Applied egg-rr 91.4%

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

10. Final simplification 91.4%

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

Alternative 5: 57.3% 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 85.5%

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

3. Simplified 79.2%

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

5. Taylor expanded in r around 0 58.7%

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

6. Final simplification 58.7%

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

Alternative 6: 13.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Simplified 79.2%

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

5. Taylor expanded in r around 0 58.7%

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

6. Taylor expanded in r around inf 16.5%

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

7. Final simplification 16.5%

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

Reproduce

herbie shell --seed 2024047 
(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))