Average Error: 52.6 → 0.2
Time: 43.0s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0692437963430577:\\ \;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\ \mathbf{elif}\;x \le 0.9687800219952365:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.6
Target45.1
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 < -1.0692437963430577

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt0.1

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

    if -1.0692437963430577 < x < 0.9687800219952365

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.9687800219952365 < x

    1. Initial program 30.6

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0692437963430577:\\ \;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\ \mathbf{elif}\;x \le 0.9687800219952365:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 
(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)))))