Average Error: 17.0 → 17.0
Time: 34.4s
Precision: 64
Internal Precision: 128
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \log \left(\left(\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*} \cdot \left(\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}}\right)\right)\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 17.0

    \[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]

  2. Taylor expanded around 0 17.0

    \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(2 \cdot x + \left(2 \cdot {x}^{2} + 1\right)\right)}\]

  3. Simplified17.0

    \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left((x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt17.0

    \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*} \cdot \sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}\right)}\]
  6. Using strategy rm

  7. Applied add-cube-cbrt17.0

    \[\leadsto \frac{1}{2} \cdot \log \left(\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right) \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right)}\right)\]

  8. Applied associate-*r*17.0

    \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\left(\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*} \cdot \left(\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right)\right) \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt17.0

    \[\leadsto \frac{1}{2} \cdot \log \left(\left(\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*} \cdot \left(\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}}\right)}\right)\]

  11. Final simplification17.0

    \[\leadsto \frac{1}{2} \cdot \log \left(\left(\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*} \cdot \left(\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{(x \cdot \left((x \cdot 2 + 2)_*\right) + 1)_*}}}\right)\right)\right)\]

Runtime

Time bar (total: 34.4s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))