Average Error: 12.1 → 1.8
Time: 3.6m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;w \le -5.62343529440539 \cdot 10^{+129}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - v \cdot 2\right)}{\frac{1 - v \cdot v}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \left(1 + v\right)\right) - 4.5\\ \mathbf{if}\;w \le -1.9611569696567568 \cdot 10^{-159}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{if}\;w \le 1.9588118475932077 \cdot 10^{-158}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;w \le 1.4800680194432159 \cdot 10^{+150}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{\sqrt[3]{\frac{1 - v \cdot v}{0.125}} \cdot \sqrt[3]{\frac{1 - v \cdot v}{0.125}}} \cdot \frac{3 - 2 \cdot v}{\sqrt[3]{\frac{1 - v \cdot v}{0.125}}}\right) \cdot \left(1 + v\right)\right) - 4.5\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 4 regimes

  2. if w < -5.62343529440539e+129

    1. Initial program 48.0

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

    3. Applied flip--48.1

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

    4. Applied associate-/r/48.1

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

    5. Applied simplify20.7

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

    7. Applied *-un-lft-identity20.7

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

    8. Applied associate-*r*20.7

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

    9. Applied simplify9.8

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

    if -5.62343529440539e+129 < w < -1.9611569696567568e-159 or 1.9588118475932077e-158 < w < 1.4800680194432159e+150

    1. Initial program 7.6

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

    3. Applied *-un-lft-identity7.6

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

    4. Applied times-frac0.5

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

    if -1.9611569696567568e-159 < w < 1.9588118475932077e-158

    1. Initial program 9.8

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

    3. Applied associate-*l*1.9

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

    if 1.4800680194432159e+150 < w

    1. Initial program 58.5

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

    3. Applied flip--58.7

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

    4. Applied associate-/r/58.7

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

    5. Applied simplify21.1

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

    7. Applied add-cube-cbrt21.2

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

    8. Applied times-frac8.2

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))