Average Error: 14.0 → 0.0
Time: 10.0s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{\frac{b}{a} \cdot \frac{b}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{\frac{b}{a} \cdot \frac{b}{a}}\right)\right|}
double f(double a, double b) {
        double r53263 = a;
        double r53264 = r53263 * r53263;
        double r53265 = b;
        double r53266 = r53265 * r53265;
        double r53267 = r53264 - r53266;
        double r53268 = r53267 / r53264;
        double r53269 = fabs(r53268);
        double r53270 = sqrt(r53269);
        return r53270;
}
↓
double f(double a, double b) {
        double r53271 = 1.0;
        double r53272 = b;
        double r53273 = a;
        double r53274 = r53272 / r53273;
        double r53275 = r53274 * r53274;
        double r53276 = exp(r53275);
        double r53277 = log(r53276);
        double r53278 = r53271 - r53277;
        double r53279 = fabs(r53278);
        double r53280 = sqrt(r53279);
        return r53280;
}

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 14.0

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

  2. Simplified14.0

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

  4. Applied times-frac0.0

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

  6. Applied add-log-exp0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))