Average Error: 59.3 → 0.1
Time: 34.6s
Precision: 64
Internal Precision: 1408
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\frac{2}{3} \cdot {x}^{3} + \left(\frac{2}{5} \cdot {x}^{5} + 2 \cdot x\right)}{2}\]

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 simplify59.3

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

  3. Taylor expanded around 0 0.1

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

Runtime

Time bar (total: 34.6s)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))