Average Error: 58.6 → 0.2
Time: 43.8s
Precision: 64
Internal Precision: 1408
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{\frac{2}{3}}{2} \cdot {x}^{3} + \left(x + \frac{\frac{2}{5}}{2} \cdot {x}^{5}\right)\]

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

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

Runtime

Time bar (total: 43.8s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))