Average Error: 52.3 → 0.1
Time: 57.4s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.081462638229255:\\ \;\;\;\;\log \left((\left(\frac{1}{x}\right) \cdot \left((\left(\frac{1}{x}\right) \cdot \left(\frac{\frac{1}{8}}{x}\right) + \left(-\frac{1}{2}\right))_*\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{if}\;x \le 0.007998961626976817:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1^2 + x^2}^* + x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.3
Target44.7
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.081462638229255

    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 -1.081462638229255 < x < 0.007998961626976817

    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 0.1

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

    if 0.007998961626976817 < x

    1. Initial program 31.0

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

    2. Applied simplify0.1

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

Runtime

Time bar (total: 57.4s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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)))))