Average Error: 0.0 → 0.0
Time: 40.8s
Precision: 64
Internal Precision: 384
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \left(\sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}} \cdot \sqrt[3]{\sqrt{\frac{\sqrt{1 - x \cdot x}}{x}}}\right)\right) \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right)\]

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-cube-cbrt0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied cbrt-prod0.0

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

Runtime

Time bar (total: 40.8s)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))