Average Error: 0.1 → 0.0
Time: 35.2s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{e^{\log_* (1 + \sqrt{1 - x \cdot x})}}{x}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

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

  2. Initial simplification0.1

    \[\leadsto \log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  3. Using strategy rm

  4. Applied div-inv0.1

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

  5. Applied distribute-rgt1-in0.1

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

  6. Applied log-prod0.2

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

  7. Simplified0.2

    \[\leadsto \color{blue}{\log_* (1 + \sqrt{1 - x \cdot x})} + \log \left(\frac{1}{x}\right)\]
  8. Using strategy rm

  9. Applied add-log-exp0.2

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

  10. Applied sum-log0.1

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

  11. Simplified0.0

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

  12. Final simplification0.0

    \[\leadsto \log \left(\frac{e^{\log_* (1 + \sqrt{1 - x \cdot x})}}{x}\right)\]

Runtime

Time bar (total: 35.2s)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))