Average Error: 52.4 → 0.1
Time: 22.9s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0454368175555075:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.005324815125416633:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{3}{40}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1^2 + x^2}^*\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.4
Target44.7
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0454368175555075

    1. Initial program 61.7

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

    2. Initial simplification60.9

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

    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

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

    if -1.0454368175555075 < x < 0.005324815125416633

    1. Initial program 58.9

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

    2. Initial simplification58.9

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

    3. Taylor expanded around 0 0.1

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

    if 0.005324815125416633 < x

    1. Initial program 30.4

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

    2. Initial simplification0.1

      \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
    3. Using strategy rm

    4. Applied *-un-lft-identity0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0454368175555075:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{-1}{16}}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.005324815125416633:\\ \;\;\;\;\left(x + {x}^{5} \cdot \frac{3}{40}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1^2 + x^2}^*\right)\\ \end{array}\]

Runtime

Time bar (total: 22.9s)Debug logProfile

herbie shell --seed 2018255 +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)))))