Average Error: 14.4 → 0.0
Time: 16.6s
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 r99305 = a;
        double r99306 = r99305 * r99305;
        double r99307 = b;
        double r99308 = r99307 * r99307;
        double r99309 = r99306 - r99308;
        double r99310 = r99309 / r99306;
        double r99311 = fabs(r99310);
        double r99312 = sqrt(r99311);
        return r99312;
}
↓
double f(double a, double b) {
        double r99313 = 1.0;
        double r99314 = b;
        double r99315 = a;
        double r99316 = r99314 / r99315;
        double r99317 = r99316 * r99316;
        double r99318 = exp(r99317);
        double r99319 = log(r99318);
        double r99320 = r99313 - r99319;
        double r99321 = fabs(r99320);
        double r99322 = sqrt(r99321);
        return r99322;
}

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 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 2020045 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))