Average Error: 52.3 → 0.2
Time: 24.4s
Precision: 64
Internal Precision: 2432
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0700427558809735:\\ \;\;\;\;\left(\frac{\frac{3}{32}}{{x}^{4}} + \log \frac{1}{2}\right) + \left(\log \left(\frac{-1}{x}\right) - \frac{\frac{\frac{1}{4}}{x}}{x}\right)\\ \mathbf{if}\;x \le 0.006949973394836614:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1^2 + x^2}^* + x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.3
Target44.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0700427558809735

    1. Initial program 61.6

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

    2. Applied simplify60.8

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

    3. Taylor expanded around -inf 0.4

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

    4. Applied simplify0.4

      \[\leadsto \color{blue}{\left(\frac{\frac{3}{32}}{{x}^{4}} + \log \frac{1}{2}\right) + \left(\log \left(\frac{-1}{x}\right) - \frac{\frac{\frac{1}{4}}{x}}{x}\right)}\]

    if -1.0700427558809735 < x < 0.006949973394836614

    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 0.006949973394836614 < 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: 24.4s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +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)))))