Average Error: 14.4 → 0.0
Time: 3.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{1 - {\left(\frac{b}{a}\right)}^{3}}{\frac{b}{a} \cdot \left(\frac{b}{a} + 1\right) + 1}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{1 - {\left(\frac{b}{a}\right)}^{3}}{\frac{b}{a} \cdot \left(\frac{b}{a} + 1\right) + 1}\right|}
double f(double a, double b) {
        double r57173 = a;
        double r57174 = r57173 * r57173;
        double r57175 = b;
        double r57176 = r57175 * r57175;
        double r57177 = r57174 - r57176;
        double r57178 = r57177 / r57174;
        double r57179 = fabs(r57178);
        double r57180 = sqrt(r57179);
        return r57180;
}
↓
double f(double a, double b) {
        double r57181 = a;
        double r57182 = b;
        double r57183 = r57181 + r57182;
        double r57184 = r57183 / r57181;
        double r57185 = 1.0;
        double r57186 = r57182 / r57181;
        double r57187 = 3.0;
        double r57188 = pow(r57186, r57187);
        double r57189 = r57185 - r57188;
        double r57190 = r57186 + r57185;
        double r57191 = r57186 * r57190;
        double r57192 = r57191 + r57185;
        double r57193 = r57189 / r57192;
        double r57194 = r57184 * r57193;
        double r57195 = fabs(r57194);
        double r57196 = sqrt(r57195);
        return r57196;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.4

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

  3. Applied difference-of-squares14.4

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

  4. Applied times-frac0.0

    \[\leadsto \sqrt{\left|\color{blue}{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right|}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube27.3

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

  7. Applied add-cbrt-cube26.7

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

  8. Applied cbrt-undiv26.7

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

  9. Simplified0.0

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

  11. Applied flip3--0.0

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

  12. Applied cube-div0.0

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

  13. Applied cbrt-div0.0

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

  14. Simplified0.0

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

  15. Simplified0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2020083 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))