Error

Target

Derivation

1. Split input into 3 regimes

2. if x < -1.0457624597562116

   1. Initial program 61.8

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

   2. Initial simplification61.0

      \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]

   3. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

   4. Simplified0.1

      \[\leadsto \log \color{blue}{\left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)}\]

   if -1.0457624597562116 < x < 1.0059846727535393

   1. Initial program 58.6

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

   2. Initial simplification58.6

      \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]

   3. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

   if 1.0059846727535393 < x

   1. Initial program 30.1

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

   2. Initial simplification0.1

      \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]

   3. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{\left(\log 2 + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\log \left(\frac{1}{x}\right) + \frac{3}{32} \cdot \frac{1}{{x}^{4}}\right)}\]

   4. Simplified0.3

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{4}}{x \cdot x} + \log 2\right) - \left(\frac{\frac{3}{32}}{{x}^{4}} - \log x\right)}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt0.7

      \[\leadsto \left(\frac{\frac{1}{4}}{x \cdot x} + \log 2\right) - \left(\frac{\frac{3}{32}}{{x}^{4}} - \color{blue}{\sqrt{\log x} \cdot \sqrt{\log x}}\right)\]

   7. Applied add-sqr-sqrt0.7

      \[\leadsto \left(\frac{\frac{1}{4}}{x \cdot x} + \log 2\right) - \left(\color{blue}{\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} \cdot \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}} - \sqrt{\log x} \cdot \sqrt{\log x}\right)\]

   8. Applied difference-of-squares0.7

      \[\leadsto \left(\frac{\frac{1}{4}}{x \cdot x} + \log 2\right) - \color{blue}{\left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right)}\]

   9. Applied add-cube-cbrt0.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{1}{4}}{x \cdot x} + \log 2} \cdot \sqrt[3]{\frac{\frac{1}{4}}{x \cdot x} + \log 2}\right) \cdot \sqrt[3]{\frac{\frac{1}{4}}{x \cdot x} + \log 2}} - \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right)\]

   10. Applied prod-diff0.7

       \[\leadsto \color{blue}{(\left(\sqrt[3]{\frac{\frac{1}{4}}{x \cdot x} + \log 2} \cdot \sqrt[3]{\frac{\frac{1}{4}}{x \cdot x} + \log 2}\right) \cdot \left(\sqrt[3]{\frac{\frac{1}{4}}{x \cdot x} + \log 2}\right) + \left(-\left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right)\right))_* + (\left(-\left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right)\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right) + \left(\left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right)\right))_*}\]

   11. Simplified0.7

       \[\leadsto \color{blue}{(\left(\sqrt{\log x} - \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}\right) \cdot \left(\sqrt{\log x} + \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}\right) + \left(\frac{\frac{\frac{1}{4}}{x}}{x} + \log 2\right))_*} + (\left(-\left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right)\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right) + \left(\left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} - \sqrt{\log x}\right) \cdot \left(\sqrt{\frac{\frac{3}{32}}{{x}^{4}}} + \sqrt{\log x}\right)\right))_*\]

   12. Simplified0.7

       \[\leadsto (\left(\sqrt{\log x} - \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}\right) \cdot \left(\sqrt{\log x} + \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}\right) + \left(\frac{\frac{\frac{1}{4}}{x}}{x} + \log 2\right))_* + \color{blue}{0}\]
3. Recombined 3 regimes into one program.

4. Final simplification0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0457624597562116:\\ \;\;\;\;\log \left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 1.0059846727535393:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\log x} - \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}\right) \cdot \left(\sqrt{\log x} + \sqrt{\frac{\frac{3}{32}}{{x}^{4}}}\right) + \left(\log 2 + \frac{\frac{\frac{1}{4}}{x}}{x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 37.8s)Debug logProfile

herbie shell --seed 2018219 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))