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Target

Derivation

1. Split input into 3 regimes

2. if x < -1.0435353126017766

   1. Initial program 61.7

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

   2. Taylor expanded around -inf 0.2

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

   3. Simplified0.2

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

   if -1.0435353126017766 < x < 0.9667467142280759

   1. Initial program 58.8

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

   2. 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 0.9667467142280759 < x

   1. Initial program 30.3

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified0.2

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\frac{-1}{8}}{x}}{x \cdot x}\right)}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0435353126017766:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9667467142280759:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{3}{40}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + \left(x + \frac{\frac{1}{2}}{x}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(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)))))

Details

Time bar (total: 17.8s)Debug log