Average Error: 52.2 → 0.2
Time: 30.5s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0700817779669067:\\ \;\;\;\;\log \left((\left(\frac{1}{x}\right) \cdot \left((\left(\frac{1}{x}\right) \cdot \left(\frac{\frac{1}{8}}{x}\right) + \left(-\frac{1}{2}\right))_*\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 0.007131525624154132:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1^2 + x^2}^* + x\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original52.2
Target44.3
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 < -1.0700817779669067

    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

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

    if -1.0700817779669067 < x < 0.007131525624154132

    1. Initial program 58.8

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

    2. Initial simplification58.8

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

    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 0.007131525624154132 < x

    1. Initial program 29.8

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

    2. Initial simplification0.1

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

    4. Applied add-cube-cbrt0.2

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

    6. Applied add-exp-log0.2

      \[\leadsto \log \color{blue}{\left(e^{\log \left(\left(\sqrt[3]{x + \sqrt{1^2 + x^2}^*} \cdot \sqrt[3]{x + \sqrt{1^2 + x^2}^*}\right) \cdot \sqrt[3]{x + \sqrt{1^2 + x^2}^*}\right)}\right)}\]

    7. Applied rem-log-exp0.2

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

    8. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0700817779669067:\\ \;\;\;\;\log \left((\left(\frac{1}{x}\right) \cdot \left((\left(\frac{1}{x}\right) \cdot \left(\frac{\frac{1}{8}}{x}\right) + \left(-\frac{1}{2}\right))_*\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 0.007131525624154132:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{1^2 + x^2}^* + x\right)\\ \end{array}\]

Runtime

Time bar (total: 30.5s)Debug logProfile

herbie shell --seed 2018230 +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)))))