Average Error: 14.2 → 0.0
Time: 15.5s
Precision: 64
Internal Precision: 384
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\left|\sqrt[3]{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right| \cdot \sqrt{\sqrt[3]{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]

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

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

  5. Applied sqrt-prod0.0

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

  6. Applied simplify0.0

    \[\leadsto \color{blue}{\left|\sqrt[3]{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\right|} \cdot \sqrt{\sqrt[3]{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]
  7. Removed slow pow expressions.

Runtime

Time bar (total: 15.5s)Debug log

herbie shell --seed '#(2961832646 520228599 1275628947 1047906571 1774476463 2890033825)' 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))