Average Error: 13.8 → 0.0
Time: 15.5s
Precision: 64
Internal Precision: 320
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{\sqrt[3]{\frac{b}{a} \cdot \frac{b}{a}} \cdot \left(\sqrt[3]{\frac{b}{a} \cdot \frac{b}{a}} \cdot \sqrt[3]{\frac{b}{a} \cdot \frac{b}{a}}\right)}\right)\right|}\]

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 13.8

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

  2. Initial simplification0.0

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

  4. Applied add-log-exp0.0

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

  6. Applied add-cube-cbrt0.0

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

  7. Final simplification0.0

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

Runtime

Time bar (total: 15.5s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.00.00.00.00%
herbie shell --seed 2018296 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))