Average Error: 14.4 → 0.0
Time: 16.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{\frac{b \cdot \frac{b}{a}}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{\frac{b \cdot \frac{b}{a}}{a}}\right)\right|}
double f(double a, double b) {
        double r118767 = a;
        double r118768 = r118767 * r118767;
        double r118769 = b;
        double r118770 = r118769 * r118769;
        double r118771 = r118768 - r118770;
        double r118772 = r118771 / r118768;
        double r118773 = fabs(r118772);
        double r118774 = sqrt(r118773);
        return r118774;
}
↓
double f(double a, double b) {
        double r118775 = 1.0;
        double r118776 = b;
        double r118777 = a;
        double r118778 = r118776 / r118777;
        double r118779 = r118776 * r118778;
        double r118780 = r118779 / r118777;
        double r118781 = exp(r118780);
        double r118782 = log(r118781);
        double r118783 = r118775 - r118782;
        double r118784 = fabs(r118783);
        double r118785 = sqrt(r118784);
        return r118785;
}

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. Simplified14.4

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

  4. Applied associate-/r*0.7

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

  5. Simplified0.0

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

  7. Applied add-log-exp0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))