Average Error: 52.6 → 0.2
Time: 52.0s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0055082235647663:\\ \;\;\;\;\left(\frac{\frac{3}{32}}{{x}^{4}} + \log \left(\frac{1}{2} \cdot \frac{-1}{x}\right)\right) - \frac{\frac{1}{4}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.008260310925820415:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left((\left(\sqrt{\sqrt{1^2 + x^2}^*}\right) \cdot \left(\sqrt{\sqrt{1^2 + x^2}^*}\right) + x)_*\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.6
Target44.8
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 < -1.0055082235647663

    1. Initial program 61.6

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

    2. Initial simplification60.8

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

    3. Taylor expanded around -inf 0.5

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

    4. 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 < 0.008260310925820415

    1. Initial program 58.8

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

    2. Initial simplification58.8

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

    3. Taylor expanded around 0 0.1

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

    if 0.008260310925820415 < x

    1. Initial program 30.9

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

    2. Initial simplification0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied fma-def0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0055082235647663:\\ \;\;\;\;\left(\frac{\frac{3}{32}}{{x}^{4}} + \log \left(\frac{1}{2} \cdot \frac{-1}{x}\right)\right) - \frac{\frac{1}{4}}{x \cdot x}\\ \mathbf{elif}\;x \le 0.008260310925820415:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left((\left(\sqrt{\sqrt{1^2 + x^2}^*}\right) \cdot \left(\sqrt{\sqrt{1^2 + x^2}^*}\right) + x)_*\right)\\ \end{array}\]

Runtime

Time bar (total: 52.0s)Debug logProfile

herbie shell --seed 2018214 +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)))))