Average Error: 17.0 → 17.0
Time: 2.4m
Precision: 64
Internal Precision: 128
\[\frac{1}{2} \cdot \log \left(\frac{1 + x}{1 - x}\right)\]
↓
\[\log \left(\sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}} \cdot \left(\left(\log \left(e^{\sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}}}\right) \cdot \sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}}\right) \cdot \sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}\right)\right) \cdot \frac{1}{2}\]

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 17.0

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

  2. Initial simplification17.0

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

  3. Taylor expanded around 0 17.0

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

  4. Simplified17.0

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

  6. Applied add-sqr-sqrt17.0

    \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)} \cdot \sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}\right)}\]
  7. Using strategy rm

  8. Applied add-cube-cbrt17.0

    \[\leadsto \frac{1}{2} \cdot \log \left(\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}} \cdot \sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}}\right) \cdot \sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}}\right)}\right)\]

  9. Applied associate-*r*17.0

    \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\left(\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)} \cdot \left(\sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}} \cdot \sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}}\right)\right) \cdot \sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(x + 1\right)}}\right)}\]
  10. Using strategy rm

  11. Applied add-log-exp17.0

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

  12. Final simplification17.0

    \[\leadsto \log \left(\sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}} \cdot \left(\left(\log \left(e^{\sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}}}\right) \cdot \sqrt[3]{\sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}}\right) \cdot \sqrt{1 + \left(x \cdot 2\right) \cdot \left(1 + x\right)}\right)\right) \cdot \frac{1}{2}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018255 
(FPCore (x)
  :name "Hyperbolic arc-(co)tangent"
  (* (/ 1 2) (log (/ (+ 1 x) (- 1 x)))))