Average Error: 52.2 → 1.7
Time: 59.5s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x} \le -5.494590308109969 \cdot 10^{-24}:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \log \left(e^{\frac{1}{6} \cdot {x}^{3}}\right)\\ \mathbf{if}\;\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x} \le 7.084021811928732 \cdot 10^{-306}:\\ \;\;\;\;\log \left(\left(\left(x + x\right) + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\\ \mathbf{if}\;\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x} \le 0.13128748121120157:\\ \;\;\;\;\log \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{3}{40} \cdot {x}^{5} + x\right) - \log \left(e^{\frac{1}{6} \cdot {x}^{3}}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.2
Target44.9
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (- (/ (/ 1/8 x) (* x x)) (/ 1/2 x)) < -5.494590308109969e-24 or 0.13128748121120157 < (- (/ (/ 1/8 x) (* x x)) (/ 1/2 x))

    1. Initial program 56.2

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

    2. Taylor expanded around 0 2.4

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

    4. Applied add-log-exp2.7

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

    if -5.494590308109969e-24 < (- (/ (/ 1/8 x) (* x x)) (/ 1/2 x)) < 7.084021811928732e-306

    1. Initial program 33.2

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

    2. Taylor expanded around inf 0.9

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

    3. Applied simplify0.9

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

    if 7.084021811928732e-306 < (- (/ (/ 1/8 x) (* x x)) (/ 1/2 x)) < 0.13128748121120157

    1. Initial program 61.9

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

    2. Taylor expanded around -inf 61.1

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

    3. Applied simplify0.2

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

Runtime

Time bar (total: 59.5s)Debug logProfile

herbie shell --seed 2018193 
(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)))))