Average Error: 12.0 → 1.6
Time: 3.2m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;w \le -1.3649845532724086 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{3 - v \cdot 2}{1 - v \cdot v} \cdot \frac{0.125 \cdot \left(w \cdot r\right)}{\frac{\frac{1}{r}}{w}} + \left(\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1} \cdot \frac{3 - 2 \cdot v}{\frac{1 - v \cdot v}{0.125}}\right) \cdot v\right)\right) - 4.5\\ \mathbf{if}\;w \le -9.46329395453615 \cdot 10^{-147}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}\right) - 4.5\\ \mathbf{if}\;w \le 2.8706814841335967 \cdot 10^{-155}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;w \le 1.1451036659984098 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{3 - v \cdot 2}{1 - v \cdot v} \cdot \frac{0.125 \cdot \left(w \cdot r\right)}{\frac{\frac{1}{r}}{w}} + \left(\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1} \cdot \frac{3 - 2 \cdot v}{\frac{1 - v \cdot v}{0.125}}\right) \cdot v\right)\right) - 4.5\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 3 regimes

  2. if w < -1.3649845532724086e+154 or 1.1451036659984098e+154 < w

    1. Initial program 61.4

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

    3. Applied flip--61.4

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

    4. Applied associate-/r/61.4

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

    5. Applied 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

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

    8. Applied times-frac8.9

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

    10. Applied distribute-lft-in8.9

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

    11. Applied simplify8.9

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

    if -1.3649845532724086e+154 < w < -9.46329395453615e-147 or 2.8706814841335967e-155 < w < 1.1451036659984098e+154

    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 associate-/l*0.4

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

    if -9.46329395453615e-147 < w < 2.8706814841335967e-155

    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*2.0

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' 
(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))