Average Error: 52.2 → 0.2
Time: 57.6s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0740806814635735:\\ \;\;\;\;\log \left((\left(\frac{1}{x}\right) \cdot \left((\left(\frac{1}{x}\right) \cdot \left(\frac{\frac{1}{8}}{x}\right) + \left(-\frac{1}{2}\right))_*\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{if}\;x \le 0.004985663317943214:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \log \left(e^{\frac{1}{6} \cdot {x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\sqrt{1^2 + x^2}^* + x}\right) + \left(\log \left(\sqrt{\sqrt{\sqrt{1^2 + x^2}^* + x}}\right) + \log \left(\sqrt{\sqrt{\sqrt{1^2 + x^2}^* + x}}\right)\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.2
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 < -1.0740806814635735

    1. Initial program 61.8

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

    2. Applied simplify61.0

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

    3. Taylor expanded around -inf 0.1

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

    4. Applied simplify0.1

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

    if -1.0740806814635735 < x < 0.004985663317943214

    1. Initial program 58.6

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

    2. Applied simplify58.6

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

    3. Taylor expanded around 0 0.1

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

    5. Applied add-log-exp0.3

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

    if 0.004985663317943214 < x

    1. Initial program 30.0

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

    2. Applied simplify0.1

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

    4. Applied add-sqr-sqrt0.2

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

    5. Applied log-prod0.2

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

    7. Applied add-sqr-sqrt0.2

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

    8. Applied sqrt-prod0.2

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

    9. Applied log-prod0.2

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

Runtime

Time bar (total: 57.6s)Debug logProfile

herbie shell --seed 2018193 +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)))))