Average Error: 14.3 → 0.0
Time: 22.1s
Precision: 64
Internal Precision: 576
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1 - {\left(\frac{b}{a}\right)}^{\left(1 + 3\right)}}{(\left(\frac{b}{a}\right) \cdot \left(\frac{b}{a}\right) + 1)_*}\right|}\]

Error

Bits error versus a

Bits error versus b

Try it out

  1. Inputs

  2. Original Output:

    Herbie Output:

Derivation

  1. Initial program 14.3

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

  2. Applied simplify0.0

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

  4. Applied flip--0.0

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

  5. Applied simplify0.0

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

  6. Applied simplify0.0

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

Runtime

Time bar (total: 22.1s)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))