Average Error: 58.6 → 0.2
Time: 2.8m
Precision: 64
Internal Precision: 1344
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[(\left(\frac{1}{2}\right) \cdot \left({x}^{5} \cdot \frac{2}{5}\right) + \left((\left({x}^{3}\right) \cdot \left(\frac{\frac{2}{3}}{2}\right) + x)_*\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 58.6

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

  2. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + \left(\frac{2}{5} \cdot {x}^{5} + 2 \cdot x\right)\right)}\]

  3. Applied simplify0.2

    \[\leadsto \color{blue}{(\left(\frac{1}{2}\right) \cdot \left({x}^{5} \cdot \frac{2}{5}\right) + \left((\left({x}^{3}\right) \cdot \left(\frac{\frac{2}{3}}{2}\right) + x)_*\right))_*}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))