Average Error: 12.5 → 0.3
Time: 4.5m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;v \le -12.361687198526626:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \left(3 - 4.5\right)\right) - \left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25 - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{v} \cdot \left(0.125 + \frac{0.125}{v}\right)\right)\\ \mathbf{if}\;v \le 24262062.621654928:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)}{\frac{\sqrt[3]{1 - v \cdot v} \cdot \sqrt[3]{1 - v \cdot v}}{\sqrt[3]{3 - \left(v + v\right)} \cdot \sqrt[3]{3 - \left(v + v\right)}} \cdot \frac{\sqrt[3]{1 - v \cdot v}}{\sqrt[3]{3 - \left(v + v\right)}}} \cdot \left(1 + v\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + \left(3 - 4.5\right)\right) - \left(\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25 - \frac{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}{v} \cdot \left(0.125 + \frac{0.125}{v}\right)\right)\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 2 regimes

  2. if v < -12.361687198526626 or 24262062.621654928 < 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. Taylor expanded around inf 17.5

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

    3. Applied simplify0.3

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

    if -12.361687198526626 < v < 24262062.621654928

    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-cube-cbrt0.4

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

    8. Applied add-cube-cbrt0.4

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

    9. Applied times-frac0.4

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

Runtime

Time bar (total: 4.5m)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))