Average Error: 52.6 → 0.0
Time: 26.4s
Precision: 64
Internal precision: 128
\[\log \left(x + \sqrt{{x}^2 + 1}\right)\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -0.20805894893085658:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{{x}^2} - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{if}\;x \le 85.36504497937187:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} + \left(x + x\right)\right) - \frac{\frac{1}{8}}{{x}^3}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.6
Comparison45.0
Herbie0.0
\[ \begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{{x}^2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{{x}^2 + 1}\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes.

  2. if x < -0.20805894893085658

    1. Initial program 62.0

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

    2. Applied taylor 60.9

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

    3. Taylor expanded around -inf 60.9

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

    4. Applied simplify 0.0

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

    5. Applied simplify 0.0

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

    if -0.20805894893085658 < x < 85.36504497937187

    1. Initial program 58.7

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

    2. Applied taylor 0.0

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

    3. Taylor expanded around 0 0.0

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

    if 85.36504497937187 < x

    1. Initial program 30.7

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

    2. Applied taylor 0.1

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

    3. Taylor expanded around inf 0.1

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

    4. Applied simplify 0.1

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

    5. Applied simplify 0.1

      \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{1}{2}}{x} + \left(x + x\right)\right) - \frac{\frac{1}{8}}{{x}^3}\right)}\]
  3. Recombined 3 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 26.4s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(303556153 1328368599 3275261765 3473155154 1077373433 3884118827)'
(FPCore (x)
  :name "Hyperbolic arcsine"

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

  (log (+ x (sqrt (+ (sqr x) 1)))))