Average Error: 13.5 → 0.0
Time: 38.5s
Precision: 64
Internal Precision: 320
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1 - {\left(\frac{b}{a} \cdot \frac{b}{a}\right)}^{3}}{\left(\frac{b}{a} \cdot \frac{b}{a} + \left(\frac{b}{a} \cdot \frac{b}{a}\right) \cdot \left(\frac{b}{a} \cdot \frac{b}{a}\right)\right) + 1}\right|}\]

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 13.5

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

  2. Initial simplification0.0

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

  4. Applied flip3--0.0

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

  5. Final simplification0.0

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

Runtime

Time bar (total: 38.5s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))