Average Error: 0.0 → 0.0
Time: 31.3s
Precision: 64
Internal Precision: 576
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \left(\sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{\sqrt{1 - x \cdot x}}{x}}\right) + \frac{1}{x}\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Final simplification0.0

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

Runtime

Time bar (total: 31.3s)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))