Average Error: 0.0 → 0.0
Time: 10.8s
Precision: 64
Internal Precision: 384
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{\sqrt{1^2 + x^2}^*}}{x}\right)\]

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 flip--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied simplify0.0

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

  6. Applied simplify0.0

    \[\leadsto \log \left(\frac{1}{x} + \frac{\frac{\sqrt{1 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{\color{blue}{\sqrt{1^2 + x^2}^*}}}{x}\right)\]

Runtime

Time bar (total: 10.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))