Average Error: 13.8 → 0.0
Time: 37.4s
Precision: 64
Internal Precision: 384
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\log \left(e^{1 - \frac{b}{a} \cdot \frac{b}{a}}\right)\right|}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 13.8

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

  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-log-exp0.0

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

Runtime

Time bar (total: 37.4s)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))