Average Error: 52.7 → 0.2
Time: 20.2s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0831908513411548:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9700681740820755:\\ \;\;\;\;\left(x + \log \left(e^{{x}^{5} \cdot \frac{3}{40}}\right)\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\frac{-1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
Target45.3
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 < -1.0831908513411548

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -1.0831908513411548 < x < 0.9700681740820755

    1. Initial program 58.6

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

    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. Using strategy rm

    4. Applied add-log-exp0.2

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

    if 0.9700681740820755 < x

    1. Initial program 31.5

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0831908513411548:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{{x}^{5}} - \left(\frac{\frac{1}{2}}{x} - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \le 0.9700681740820755:\\ \;\;\;\;\left(x + \log \left(e^{{x}^{5} \cdot \frac{3}{40}}\right)\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{\frac{1}{2}}{x}\right) + \frac{\frac{\frac{-1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 20.2s)Debug logProfile

herbie shell --seed 2018273 
(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)))))