Average Error: 58.5 → 0.0
Time: 6.4m
Precision: 64
Internal Precision: 1344
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\frac{1}{2} \cdot \log_* (1 + (e^{\left(\log_* (1 + x) + \log_* (1 + x)\right) - \left(\log \left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) - \log_* (1 + (\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(x \cdot x\right))_*)\right)} - 1)^*)\]

Error

Bits error versus x

Derivation

  1. Initial program 58.5

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

  3. Applied flip--58.6

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

  4. Applied associate-/r/58.6

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

  5. Applied log-prod58.6

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

  6. Applied simplify58.6

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

  7. Applied simplify50.5

    \[\leadsto \frac{1}{2} \cdot \left(\log \left(\frac{x + 1}{1 - x \cdot x}\right) + \color{blue}{\log_* (1 + x)}\right)\]
  8. Using strategy rm

  9. Applied log1p-expm1-u50.5

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\log_* (1 + (e^{\log \left(\frac{x + 1}{1 - x \cdot x}\right) + \log_* (1 + x)} - 1)^*)}\]

  10. Applied simplify0.5

    \[\leadsto \frac{1}{2} \cdot \log_* (1 + \color{blue}{(e^{\left(\log_* (1 + x) + \log_* (1 + x)\right) - \log \left(1 - x \cdot x\right)} - 1)^*})\]
  11. Using strategy rm

  12. Applied flip3--0.6

    \[\leadsto \frac{1}{2} \cdot \log_* (1 + (e^{\left(\log_* (1 + x) + \log_* (1 + x)\right) - \log \color{blue}{\left(\frac{{1}^{3} - {\left(x \cdot x\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)}\right)}} - 1)^*)\]

  13. Applied log-div0.6

    \[\leadsto \frac{1}{2} \cdot \log_* (1 + (e^{\left(\log_* (1 + x) + \log_* (1 + x)\right) - \color{blue}{\left(\log \left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right) + 1 \cdot \left(x \cdot x\right)\right)\right)\right)}} - 1)^*)\]

  14. Applied simplify0.0

    \[\leadsto \frac{1}{2} \cdot \log_* (1 + (e^{\left(\log_* (1 + x) + \log_* (1 + x)\right) - \left(\log \left({1}^{3} - {\left(x \cdot x\right)}^{3}\right) - \color{blue}{\log_* (1 + (\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(x \cdot x\right))_*)}\right)} - 1)^*)\]

Runtime

Time bar (total: 6.4m)Debug logProfile

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