Average Error: 0.0 → 0.0
Time: 17.1s
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left((\left(\frac{1}{\sqrt{x}}\right) \cdot \left(\frac{1}{\sqrt{x}}\right) + \left(\frac{\sqrt{1 - x \cdot x}}{x}\right))_*\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 0.0

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

  3. Applied add-sqr-sqrt0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Applied fma-def0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019091 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))