Average Error: 58.5 → 0.2
Time: 6.4m
Precision: 64
Internal Precision: 1344
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[x + \left({x}^{3} \cdot \frac{\frac{2}{3}}{2} + \frac{{x}^{5} \cdot \frac{2}{5}}{2}\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 58.5

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

  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

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

  5. Applied associate-+l+0.2

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

Runtime

Time bar (total: 6.4m)Debug logProfile

herbie shell --seed 2018199 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))