Average Error: 52.0 → 0.2
Time: 40.2s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0570275512575042:\\ \;\;\;\;\log \left((\left(\frac{1}{x}\right) \cdot \left((\left(\frac{1}{x}\right) \cdot \left(\frac{\frac{1}{8}}{x}\right) + \left(-\frac{1}{2}\right))_*\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{if}\;x \le 1.025052196293905:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{1}{4}}{x}}{x} - \frac{\frac{3}{32}}{{x}^{4}}\right) + \left(\log x + \log 2\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.0
Target44.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0570275512575042

    1. Initial program 61.7

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

    2. Applied simplify60.9

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

    3. Taylor expanded around -inf 0.2

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

    4. Applied simplify0.2

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

    if -1.0570275512575042 < x < 1.025052196293905

    1. Initial program 58.7

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

    2. Applied simplify58.7

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

    3. Taylor expanded around 0 0.2

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

    if 1.025052196293905 < x

    1. Initial program 29.5

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

    2. Applied simplify0.0

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

    3. Taylor expanded around inf 0.4

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

    4. Applied simplify0.4

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

Runtime

Time bar (total: 40.2s)Debug logProfile

herbie shell --seed 2018199 +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)))))