Average Error: 12.1 → 2.5
Time: 3.0m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;w \le -1.473123078256138 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1} \cdot \frac{3 - 2 \cdot v}{\frac{1 - v \cdot v}{0.125}}\right) \cdot \left(1 + v\right)\right) - 4.5\\ \mathbf{if}\;w \le -2.0980066480848485 \cdot 10^{-143}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{if}\;w \le 3.8907796740587485 \cdot 10^{-172}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;w \le 3.853983191199187 \cdot 10^{+145}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1} \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(3 \cdot 3 - \left(2 \cdot v\right) \cdot \left(2 \cdot v\right)\right)}{\frac{1 - v \cdot v}{0.125} \cdot \left(3 + 2 \cdot v\right)} \cdot \left(1 + v\right)\right) - 4.5\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 4 regimes

  2. if w < -1.473123078256138e+154

    1. Initial program 61.5

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

    3. Applied flip--61.5

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

    4. Applied associate-/r/61.5

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

    5. Applied simplify21.6

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

    7. Applied *-un-lft-identity21.6

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

    8. Applied times-frac9.7

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

    if -1.473123078256138e+154 < w < -2.0980066480848485e-143 or 3.8907796740587485e-172 < w < 3.853983191199187e+145

    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 times-frac1.0

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

    if -2.0980066480848485e-143 < w < 3.8907796740587485e-172

    1. Initial program 9.7

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

    3. Applied associate-*l*1.8

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

    if 3.853983191199187e+145 < w

    1. Initial program 55.5

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

    3. Applied flip--55.4

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

    4. Applied associate-/r/55.4

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

    5. Applied simplify19.4

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

    7. Applied flip--22.1

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

    8. Applied associate-*r/26.1

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

    9. Applied associate-/l/26.1

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(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))