Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1 + \sqrt{1 - x \cdot x}}{x}\right)\]

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

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

  4. Applied associate-/r*0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied associate-/r*0.0

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

  7. Applied frac-add0.0

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))