Average Error: 13.7 → 0.0
Time: 4.4s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\log \left(e^{\sqrt{\left|\left(\frac{a + b}{a} \cdot \frac{\sqrt{a - b}}{\sqrt{a}}\right) \cdot \frac{\sqrt{a - b}}{\sqrt{a}}\right|}}\right)\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\log \left(e^{\sqrt{\left|\left(\frac{a + b}{a} \cdot \frac{\sqrt{a - b}}{\sqrt{a}}\right) \cdot \frac{\sqrt{a - b}}{\sqrt{a}}\right|}}\right)
double f(double a, double b) {
        double r70713 = a;
        double r70714 = r70713 * r70713;
        double r70715 = b;
        double r70716 = r70715 * r70715;
        double r70717 = r70714 - r70716;
        double r70718 = r70717 / r70714;
        double r70719 = fabs(r70718);
        double r70720 = sqrt(r70719);
        return r70720;
}
↓
double f(double a, double b) {
        double r70721 = a;
        double r70722 = b;
        double r70723 = r70721 + r70722;
        double r70724 = r70723 / r70721;
        double r70725 = r70721 - r70722;
        double r70726 = sqrt(r70725);
        double r70727 = sqrt(r70721);
        double r70728 = r70726 / r70727;
        double r70729 = r70724 * r70728;
        double r70730 = r70729 * r70728;
        double r70731 = fabs(r70730);
        double r70732 = sqrt(r70731);
        double r70733 = exp(r70732);
        double r70734 = log(r70733);
        return r70734;
}

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 13.7

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

  3. Applied difference-of-squares13.7

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

  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-log-exp0.0

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

  8. Applied add-sqr-sqrt0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied times-frac0.0

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

  11. Applied associate-*r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))