Average Error: 59.3 → 0.1
Time: 19.3s
Precision: 64
Internal precision: 128
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
⬇
\[\frac{\frac{2}{3} \cdot {x}^3 + \left({x}^{5} \cdot \frac{2}{5} + \left(x + x\right)\right)}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 59.3

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

  2. Applied simplify 59.3

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

  3. Applied taylor 0.1

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

  4. Taylor expanded around 0 0.1

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

  5. Applied simplify 0.1

    \[\leadsto \color{blue}{\frac{\frac{2}{3} \cdot {x}^3 + \left({x}^{5} \cdot \frac{2}{5} + \left(x + x\right)\right)}{2}}\]
  6. Removed slow pow expressions

Runtime

Time bar (total: 19.3s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(1004939270 1667071945 1172085305 3394829916 2060820384 2218463993)'
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))