Average Error: 52.6 → 0.3
Time: 1.1m
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0055082235647663:\\ \;\;\;\;\left(\frac{\frac{3}{32}}{{x}^{4}} + \log \left(\frac{1}{2} \cdot \frac{-1}{x}\right)\right) - \frac{\frac{1}{4}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.954764076611607:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} + x\right) - \frac{\frac{1}{8}}{{x}^{3}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target44.8
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0055082235647663

    1. Initial program 61.6

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

    2. Taylor expanded around -inf 0.5

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

    3. Simplified0.3

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

    if -1.0055082235647663 < x < 0.954764076611607

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    if 0.954764076611607 < x

    1. Initial program 31.1

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0055082235647663:\\ \;\;\;\;\left(\frac{\frac{3}{32}}{{x}^{4}} + \log \left(\frac{1}{2} \cdot \frac{-1}{x}\right)\right) - \frac{\frac{1}{4}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.954764076611607:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} + x\right) - \frac{\frac{1}{8}}{{x}^{3}}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018214 
(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)))))