Average Error: 63.0 → 63.0
Time: 4.3m
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(e^{\log \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right)} + \frac{1}{x}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 63.0

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

  3. Applied add-exp-log63.0

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

  5. Applied add-cube-cbrt63.0

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

  6. Applied log-prod63.0

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

  7. Final simplification63.0

    \[\leadsto \log \left(e^{\log \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right)} + \frac{1}{x}\right)\]

Runtime

Time bar (total: 4.3m)Debug logProfile

herbie shell --seed 2018255 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))