Average Error: 14.2 → 0.0
Time: 44.0s
Precision: 64
Internal Precision: 320
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \left(\frac{b}{a} \cdot \left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right)\right) \cdot \sqrt[3]{\frac{b}{a}}\right|}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.2

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

  2. Applied simplify0.0

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

  4. Applied add-cube-cbrt0.0

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right) \cdot \sqrt[3]{\frac{b}{a}}\right)}\right|}\]

  5. Applied associate-*r*0.0

    \[\leadsto \sqrt{\left|1 - \color{blue}{\left(\frac{b}{a} \cdot \left(\sqrt[3]{\frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a}}\right)\right) \cdot \sqrt[3]{\frac{b}{a}}}\right|}\]

Runtime

Time bar (total: 44.0s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))