Average Error: 13.7 → 0.0
Time: 34.6s
Precision: 64
Internal Precision: 384
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - e^{\log \left(\frac{b}{a}\right) + \log \left(\frac{b}{a}\right)}\right|}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 13.7

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

  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-exp-log0.0

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

  5. Applied add-exp-log0.0

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

  6. Applied prod-exp0.0

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

Runtime

Time bar (total: 34.6s)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))