Average Error: 12.2 → 0.6
Time: 52.5s
Precision: 64
Internal Precision: 576
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{2}{r \cdot r}\right) - (\left(\sqrt[3]{\left((e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^* \cdot (e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^*\right) \cdot (e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^*}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.2

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

  2. Initial simplification0.4

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

  4. Applied add-cbrt-cube6.9

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

  5. Applied add-cbrt-cube20.9

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

  6. Applied cbrt-undiv20.9

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

  7. Simplified0.5

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

  9. Applied expm1-log1p-u0.5

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - (\color{blue}{\left((e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^*\right)} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) + 4.5)_*\]
  10. Using strategy rm

  11. Applied add-cbrt-cube0.6

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

  12. Final simplification0.6

    \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - (\left(\sqrt[3]{\left((e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^* \cdot (e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^*\right) \cdot (e^{\log_* (1 + \sqrt[3]{{\left(\frac{0.125}{\frac{1 - v}{(v \cdot -2 + 3)_*}}\right)}^{3}})} - 1)^*}\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) + 4.5)_*\]

Runtime

Time bar (total: 52.5s)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(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))