Average Error: 12.5 → 0.4
Time: 4.6m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;v \le -3.295616946978752:\\ \;\;\;\;\left(\left(\frac{\frac{2}{r}}{r} + \left(3 - 4.5\right)\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) + \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{v} \cdot \left(0.125 + \frac{0.125}{v}\right)\\ \mathbf{if}\;v \le 1.4461390885912786:\\ \;\;\;\;\left(\frac{2}{r \cdot r} - \frac{0.125 \cdot v + 0.125}{\left(\frac{1}{3} + \frac{2}{9} \cdot v\right) - \left(v \cdot v\right) \cdot \frac{5}{27}} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) + \left(3 - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\frac{2}{r}}{r} + \left(3 - 4.5\right)\right) - 0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) + \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{v} \cdot \left(0.125 + \frac{0.125}{v}\right)\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 2 regimes

  2. if v < -3.295616946978752 or 1.4461390885912786 < v

    1. Initial program 16.8

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

    3. Applied flip--18.5

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

    4. Applied associate-/r/18.5

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

    5. Applied simplify11.3

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

    7. Applied add-sqr-sqrt11.4

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

    8. Taylor expanded around inf 17.6

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

    9. Applied simplify0.3

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

    if -3.295616946978752 < v < 1.4461390885912786

    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 flip--8.3

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

    4. Applied associate-/r/8.3

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

    5. Applied simplify0.3

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

    7. Applied add-sqr-sqrt0.5

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

    8. Taylor expanded around 0 0.7

      \[\leadsto \left(\left(3 + \sqrt{\frac{2}{r \cdot r}} \cdot \sqrt{\frac{2}{r \cdot r}}\right) - \frac{0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{\color{blue}{\left(\frac{1}{3} + \frac{2}{9} \cdot v\right) - \frac{5}{27} \cdot {v}^{2}}} \cdot \left(1 + v\right)\right) - 4.5\]

    9. Applied simplify0.5

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} - \frac{0.125 \cdot v + 0.125}{\left(\frac{1}{3} + \frac{2}{9} \cdot v\right) - \left(v \cdot v\right) \cdot \frac{5}{27}} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right) + \left(3 - 4.5\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))