Average Error: 14.0 → 0.0
Time: 30.2s
Precision: 64
Internal Precision: 320
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\log_* (1 + \log \left(e^{(e^{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} - 1)^*}\right))\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.0

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Applied simplify0.0

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]
  3. Using strategy rm

  4. Applied log1p-expm1-u0.0

    \[\leadsto \color{blue}{\log_* (1 + (e^{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} - 1)^*)}\]
  5. Using strategy rm

  6. Applied add-log-exp0.0

    \[\leadsto \log_* (1 + \color{blue}{\log \left(e^{(e^{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} - 1)^*}\right)})\]

Runtime

Time bar (total: 30.2s)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))