Average Error: 12.3 → 0.6
Time: 3.3m
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;v \le -1.3554637334972328 \cdot 10^{+29} \lor \neg \left(v \le 130860.82334818215\right):\\ \;\;\;\;\left(\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\right) - \left(\sqrt[3]{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - 2 \cdot v\right)} \cdot \sqrt[3]{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - 2 \cdot v\right)}\right) \cdot \sqrt[3]{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25 - \frac{w \cdot r}{\frac{v}{w \cdot r}} \cdot \left(0.125 + \frac{0.125}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left({3}^{3} - {\left(2 \cdot v\right)}^{3}\right)\right) \cdot 0.125}{\left(1 - v\right) \cdot \left(3 \cdot 3 + \left(3 \cdot \left(2 \cdot v\right) + \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)\right)}\right) - 4.5\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if v < -1.3554637334972328e+29 or 130860.82334818215 < v

    1. Initial program 16.9

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

    3. Applied *-un-lft-identity16.9

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

    4. Applied associate-/r*16.9

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

    5. Applied simplify12.9

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

    7. Applied add-cube-cbrt13.0

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

    8. Taylor expanded around inf 22.9

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\sqrt[3]{\frac{0.125 \cdot \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(3 - 2 \cdot v\right)\right)}{1 - v}} \cdot \sqrt[3]{\frac{0.125 \cdot \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(3 - 2 \cdot v\right)\right)}{1 - v}}\right) \cdot \sqrt[3]{\color{blue}{0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right) - \left(0.125 \cdot \frac{{w}^{2} \cdot {r}^{2}}{v} + 0.125 \cdot \frac{{w}^{2} \cdot {r}^{2}}{{v}^{2}}\right)}}\right) - 4.5\]

    9. Applied simplify0.6

      \[\leadsto \color{blue}{\left(\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\right) - \sqrt[3]{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25 - \frac{w \cdot r}{\frac{v}{w \cdot r}} \cdot \left(\frac{0.125}{v} + 0.125\right)} \cdot \left(\sqrt[3]{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - v \cdot 2\right)} \cdot \sqrt[3]{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - v \cdot 2\right)}\right)}\]

    if -1.3554637334972328e+29 < v < 130860.82334818215

    1. Initial program 8.2

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

    3. Applied *-un-lft-identity8.2

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

    4. Applied associate-/r*8.2

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

    5. Applied simplify0.4

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

    7. Applied flip3--0.4

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

    8. Applied associate-*r/0.6

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

    9. Applied associate-*r/0.6

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

    10. Applied associate-/l/0.6

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

  4. Applied simplify0.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;v \le -1.3554637334972328 \cdot 10^{+29} \lor \neg \left(v \le 130860.82334818215\right):\\ \;\;\;\;\left(\left(\frac{\frac{2}{r}}{r} + 3\right) - 4.5\right) - \left(\sqrt[3]{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - 2 \cdot v\right)} \cdot \sqrt[3]{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - 2 \cdot v\right)}\right) \cdot \sqrt[3]{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25 - \frac{w \cdot r}{\frac{v}{w \cdot r}} \cdot \left(0.125 + \frac{0.125}{v}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \frac{\left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot \left({3}^{3} - {\left(2 \cdot v\right)}^{3}\right)\right) \cdot 0.125}{\left(1 - v\right) \cdot \left(3 \cdot 3 + \left(3 \cdot \left(2 \cdot v\right) + \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)\right)}\right) - 4.5\\ \end{array}}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018201 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))