Average Error: 14.1 → 0.0
Time: 23.5s
Precision: 64
Internal Precision: 320
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1 - \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(\frac{b}{a} \cdot \frac{b}{a}\right)}{\frac{b}{a} \cdot \frac{b}{a} + 1}\right|}\]

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.1

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

  2. Initial simplification0.0

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
  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. Final simplification0.0

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

Runtime

Time bar (total: 23.5s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))