Average Error: 52.2 → 0.3
Time: 25.2s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.037471742873375:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 1.0190645183237865:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{4}}{x \cdot x} - \frac{\frac{3}{32}}{{x}^{4}}\right) + \frac{{\left(\log 2\right)}^{3} + {\left(\log x\right)}^{3}}{\left(\log x \cdot \log x - \log x \cdot \log 2\right) + \log 2 \cdot \log 2}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.2
Target44.7
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 < -1.037471742873375

    1. Initial program 61.8

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

    2. Initial simplification61.8

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

    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. Simplified0.2

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

    if -1.037471742873375 < x < 1.0190645183237865

    1. Initial program 58.7

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

    2. Initial simplification58.7

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

    3. Taylor expanded around 0 0.2

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

    if 1.0190645183237865 < x

    1. Initial program 30.8

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

    2. Initial simplification30.8

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

    3. Taylor expanded around inf 0.4

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

    4. Simplified0.4

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

    6. Applied flip3-+0.8

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.037471742873375:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 1.0190645183237865:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{4}}{x \cdot x} - \frac{\frac{3}{32}}{{x}^{4}}\right) + \frac{{\left(\log 2\right)}^{3} + {\left(\log x\right)}^{3}}{\left(\log x \cdot \log x - \log x \cdot \log 2\right) + \log 2 \cdot \log 2}\\ \end{array}\]

Runtime

Time bar (total: 25.2s)Debug logProfile

herbie shell --seed 2018234 
(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)))))