Average Error: 58.4 → 0.2
Time: 28.3s
Precision: 64
Internal Precision: 1408
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\left(\frac{{x}^{5}}{\frac{2}{\frac{2}{5}}} + \frac{{x}^{3}}{\frac{2}{\frac{2}{3}}}\right) + x\]

Error

Bits error versus x

Derivation

  1. Initial program 58.4

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

  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

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

Runtime

Time bar (total: 28.3s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))