Average Error: 53.1 → 0.0
Time: 10.2s
Precision: 64
Internal precision: 2432
\[\log \left(x + \sqrt{{x}^2 + 1}\right)\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -0.11215751184115397:\\ \;\;\;\;\log \left(\frac{\frac{1}{8}}{{x}^3} - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{if}\;x \le 7.988163388129748:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + x\right) + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{{x}^3}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original53.1
Comparison45.7
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 < -0.11215751184115397

    1. Initial program 62.0

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

    2. Applied taylor 61.0

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

    3. Taylor expanded around -inf 61.0

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

    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

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

    if -0.11215751184115397 < x < 7.988163388129748

    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 7.988163388129748 < x

    1. Initial program 33.1

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

    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

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

    5. Applied simplify 0.1

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

Runtime

Time bar (total: 10.2s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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)))))