Average Error: 59.2 → 0.1
Time: 45.5s
Precision: 64
Internal Precision: 1408
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\left(\frac{2}{3} \cdot {x}^{3} + \frac{2}{5} \cdot {x}^{5}\right) + 2 \cdot x}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 59.2

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

  2. Applied simplify59.2

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

  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. Using strategy rm

  5. Applied associate-+r+0.1

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

Runtime

Time bar (total: 45.5s)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))