Average Error: 52.4 → 0.3
Time: 17.9s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.045195896304336:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 1.009821353901925:\\ \;\;\;\;(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot x + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(\frac{\frac{-3}{32}}{{x}^{4}} + \left(\frac{\frac{1}{4}}{x \cdot x} + \log x\right)\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.4
Target44.8
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.045195896304336

    1. Initial program 61.7

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

    2. Simplified60.9

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

    3. Taylor expanded around -inf 0.3

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

    4. Simplified0.3

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

    if -1.045195896304336 < x < 1.009821353901925

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

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

    if 1.009821353901925 < x

    1. Initial program 30.9

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

    2. Simplified0.1

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.045195896304336:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 1.009821353901925:\\ \;\;\;\;(\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right) \cdot x + \left((\frac{3}{40} \cdot \left({x}^{5}\right) + x)_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(\frac{\frac{-3}{32}}{{x}^{4}} + \left(\frac{\frac{1}{4}}{x \cdot x} + \log x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +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)))))