Average Error: 52.4 → 0.2
Time: 38.7s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9966636267464427:\\ \;\;\;\;\log \left(\left(-\frac{\frac{1}{2}}{x}\right) \cdot e^{\frac{\frac{3}{32}}{{x}^{4}} - \frac{\frac{\frac{1}{4}}{x}}{x}}\right)\\ \mathbf{elif}\;x \le 0.9616571784347575:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} + x\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.4
Target44.9
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 < -0.9966636267464427

    1. Initial program 61.7

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

    2. Taylor expanded around -inf 0.5

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

    3. Simplified0.5

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

    5. Applied add-log-exp0.5

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

    6. Applied diff-log0.5

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

    7. Applied add-log-exp0.5

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

    8. Applied sum-log0.3

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

    9. Simplified0.3

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

    if -0.9966636267464427 < x < 0.9616571784347575

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.2

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

    if 0.9616571784347575 < x

    1. Initial program 30.7

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9966636267464427:\\ \;\;\;\;\log \left(\left(-\frac{\frac{1}{2}}{x}\right) \cdot e^{\frac{\frac{3}{32}}{{x}^{4}} - \frac{\frac{\frac{1}{4}}{x}}{x}}\right)\\ \mathbf{elif}\;x \le 0.9616571784347575:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(\frac{\frac{1}{2}}{x} + x\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 38.7s)Debug logProfile

herbie shell --seed 2018225 
(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)))))