Average Error: 12.5 → 0.5
Time: 3.0m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;v \le -1.2674889682837169 \cdot 10^{+46}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - 4.5\right) - \sqrt{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - v \cdot 2\right)} \cdot \sqrt{\left(0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right) - \frac{w \cdot r}{\frac{v}{w \cdot r}} \cdot \left(0.125 + \frac{0.125}{v}\right)}\\ \mathbf{if}\;v \le 3278914316.594352:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(3 \cdot 3 - \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)\right)}{\left(1 - v\right) \cdot \left(3 + 2 \cdot v\right)}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - 4.5\right) - \sqrt{\left(\frac{0.125}{1 - v} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) \cdot \left(3 - v \cdot 2\right)} \cdot \sqrt{\left(0.25 \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right) - \frac{w \cdot r}{\frac{v}{w \cdot r}} \cdot \left(0.125 + \frac{0.125}{v}\right)}\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 2 regimes

  2. if v < -1.2674889682837169e+46 or 3278914316.594352 < v

    1. Initial program 17.3

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

    3. Applied *-un-lft-identity17.3

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

    4. Applied associate-/r*17.3

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

    5. Applied simplify13.7

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

    7. Applied add-sqr-sqrt13.7

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

    8. Taylor expanded around inf 23.3

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

    9. Applied simplify0.4

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

    if -1.2674889682837169e+46 < v < 3278914316.594352

    1. Initial program 8.3

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

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

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

    4. Applied associate-/r*8.3

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

    5. Applied simplify0.5

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

    7. Applied flip--0.5

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

    8. Applied associate-*r/0.6

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

    9. Applied associate-*r/0.6

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

    10. Applied associate-/l/0.6

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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))