Eccentricity of an ellipse

1. Initial program 76.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Step-by-step derivation

   1. div-sub76.9%

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

   2. *-inverses76.9%

      \[\leadsto \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]

   3. times-frac100.0%

      \[\leadsto \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} \]

4. Taylor expanded in b around 0 76.9%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{{b}^{2}}{{a}^{2}}\right|}} \]
5. Step-by-step derivation

   1. fabs-sub76.9%

      \[\leadsto \sqrt{\color{blue}{\left|\frac{{b}^{2}}{{a}^{2}} - 1\right|}} \]

   2. unpow276.9%

      \[\leadsto \sqrt{\left|\frac{\color{blue}{b \cdot b}}{{a}^{2}} - 1\right|} \]

   3. unpow276.9%

      \[\leadsto \sqrt{\left|\frac{b \cdot b}{\color{blue}{a \cdot a}} - 1\right|} \]

   4. times-frac100.0%

      \[\leadsto \sqrt{\left|\color{blue}{\frac{b}{a} \cdot \frac{b}{a}} - 1\right|} \]

   5. unpow2100.0%

      \[\leadsto \sqrt{\left|\color{blue}{{\left(\frac{b}{a}\right)}^{2}} - 1\right|} \]

   6. fabs-sub100.0%

      \[\leadsto \sqrt{\color{blue}{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}} \]

   7. rem-square-sqrt100.0%

      \[\leadsto \sqrt{\left|\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}\right|} \]

   8. fabs-sqr100.0%

      \[\leadsto \sqrt{\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}} \]

   9. rem-square-sqrt100.0%

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

6. Simplified100.0%

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}} \]
7. Step-by-step derivation

   1. unpow2100.0%

      \[\leadsto \sqrt{1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}} \]

   2. clear-num100.0%

      \[\leadsto \sqrt{1 - \frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}} \]

   3. un-div-inv100.0%

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}} \]

8. Applied egg-rr100.0%

   \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}}} \]

9. Final simplification100.0%

   \[\leadsto \sqrt{1 - \frac{\frac{b}{a}}{\frac{a}{b}}} \]

1. Initial program 76.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Step-by-step derivation

   1. div-sub76.9%

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

   2. *-inverses76.9%

      \[\leadsto \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]

   3. times-frac100.0%

      \[\leadsto \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} \]

4. Taylor expanded in b around 0 76.9%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{{b}^{2}}{{a}^{2}}\right|}} \]
5. Step-by-step derivation

   1. fabs-sub76.9%

      \[\leadsto \sqrt{\color{blue}{\left|\frac{{b}^{2}}{{a}^{2}} - 1\right|}} \]

   2. unpow276.9%

      \[\leadsto \sqrt{\left|\frac{\color{blue}{b \cdot b}}{{a}^{2}} - 1\right|} \]

   3. unpow276.9%

      \[\leadsto \sqrt{\left|\frac{b \cdot b}{\color{blue}{a \cdot a}} - 1\right|} \]

   4. times-frac100.0%

      \[\leadsto \sqrt{\left|\color{blue}{\frac{b}{a} \cdot \frac{b}{a}} - 1\right|} \]

   5. unpow2100.0%

      \[\leadsto \sqrt{\left|\color{blue}{{\left(\frac{b}{a}\right)}^{2}} - 1\right|} \]

   6. fabs-sub100.0%

      \[\leadsto \sqrt{\color{blue}{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}} \]

   7. rem-square-sqrt100.0%

      \[\leadsto \sqrt{\left|\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}\right|} \]

   8. fabs-sqr100.0%

      \[\leadsto \sqrt{\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}} \]

   9. rem-square-sqrt100.0%

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

6. Simplified100.0%

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

7. Taylor expanded in b around 0 76.5%

   \[\leadsto \color{blue}{1 + -0.5 \cdot \frac{{b}^{2}}{{a}^{2}}} \]
8. Step-by-step derivation

   1. unpow276.5%

      \[\leadsto 1 + -0.5 \cdot \frac{\color{blue}{b \cdot b}}{{a}^{2}} \]

   2. unpow276.5%

      \[\leadsto 1 + -0.5 \cdot \frac{b \cdot b}{\color{blue}{a \cdot a}} \]

   3. times-frac99.3%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)} \]

   4. unpow299.3%

      \[\leadsto 1 + -0.5 \cdot \color{blue}{{\left(\frac{b}{a}\right)}^{2}} \]

9. Simplified99.3%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {\left(\frac{b}{a}\right)}^{2}} \]
10. Step-by-step derivation

    1. *-commutative99.3%

       \[\leadsto 1 + \color{blue}{{\left(\frac{b}{a}\right)}^{2} \cdot -0.5} \]

    2. unpow299.3%

       \[\leadsto 1 + \color{blue}{\left(\frac{b}{a} \cdot \frac{b}{a}\right)} \cdot -0.5 \]

    3. clear-num99.3%

       \[\leadsto 1 + \left(\frac{b}{a} \cdot \color{blue}{\frac{1}{\frac{a}{b}}}\right) \cdot -0.5 \]

    4. div-inv99.3%

       \[\leadsto 1 + \color{blue}{\frac{\frac{b}{a}}{\frac{a}{b}}} \cdot -0.5 \]

    5. associate-*l/99.3%

       \[\leadsto 1 + \color{blue}{\frac{\frac{b}{a} \cdot -0.5}{\frac{a}{b}}} \]

11. Applied egg-rr99.3%

    \[\leadsto 1 + \color{blue}{\frac{\frac{b}{a} \cdot -0.5}{\frac{a}{b}}} \]

12. Final simplification99.3%

    \[\leadsto 1 + \frac{\frac{b}{a} \cdot -0.5}{\frac{a}{b}} \]

1. Initial program 76.9%

   \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|} \]
2. Step-by-step derivation

   1. div-sub76.9%

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

   2. *-inverses76.9%

      \[\leadsto \sqrt{\left|\color{blue}{1} - \frac{b \cdot b}{a \cdot a}\right|} \]

   3. times-frac100.0%

      \[\leadsto \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|} \]

3. Simplified100.0%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}} \]

4. Taylor expanded in b around 0 76.9%

   \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{{b}^{2}}{{a}^{2}}\right|}} \]
5. Step-by-step derivation

   1. fabs-sub76.9%

      \[\leadsto \sqrt{\color{blue}{\left|\frac{{b}^{2}}{{a}^{2}} - 1\right|}} \]

   2. unpow276.9%

      \[\leadsto \sqrt{\left|\frac{\color{blue}{b \cdot b}}{{a}^{2}} - 1\right|} \]

   3. unpow276.9%

      \[\leadsto \sqrt{\left|\frac{b \cdot b}{\color{blue}{a \cdot a}} - 1\right|} \]

   4. times-frac100.0%

      \[\leadsto \sqrt{\left|\color{blue}{\frac{b}{a} \cdot \frac{b}{a}} - 1\right|} \]

   5. unpow2100.0%

      \[\leadsto \sqrt{\left|\color{blue}{{\left(\frac{b}{a}\right)}^{2}} - 1\right|} \]

   6. fabs-sub100.0%

      \[\leadsto \sqrt{\color{blue}{\left|1 - {\left(\frac{b}{a}\right)}^{2}\right|}} \]

   7. rem-square-sqrt100.0%

      \[\leadsto \sqrt{\left|\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}\right|} \]

   8. fabs-sqr100.0%

      \[\leadsto \sqrt{\color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}} \cdot \sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}}} \]

   9. rem-square-sqrt100.0%

      \[\leadsto \sqrt{\color{blue}{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

6. Simplified100.0%

   \[\leadsto \color{blue}{\sqrt{1 - {\left(\frac{b}{a}\right)}^{2}}} \]

7. Taylor expanded in b around 0 97.8%

   \[\leadsto \color{blue}{1} \]

8. Final simplification97.8%

   \[\leadsto 1 \]

herbie shell --seed 2023166 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (and (and (<= 0.0 b) (<= b a)) (<= a 1.0))
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))