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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--58.5

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

4. Applied associate-/r/58.5

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

5. Applied log-prod58.5

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

6. Applied simplify58.5

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

7. Applied simplify50.4

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

9. Applied log-div50.4

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

10. Applied simplify0.6

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

12. Applied log1p-expm1-u0.6

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

13. Applied simplify0.0

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

Runtime

Time bar (total: 6.1m)Debug logProfile

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))