Average Error: 58.7 → 0.2
Time: 8.0s
Precision: 64
Internal precision: 1408
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
⬇
\[\frac{\left(x + x\right) + \left({x}^3 \cdot \frac{2}{3} + \log \left(e^{\frac{2}{5} \cdot {x}^{5}}\right)\right)}{2}\]

Error

Bits error versus x

Derivation

  1. Initial program 58.7

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

  2. Applied simplify 58.7

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

  3. Applied taylor 0.0

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

  4. Taylor expanded around 0 0.0

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

  5. Applied simplify 0.0

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

  7. Applied add-log-exp 0.2

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

Runtime

Time bar (total: 8.0s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))