Average Error: 14.2 → 0.0
Time: 4.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\log \left(e^{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\log \left(e^{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right)\right|}
double f(double a, double b) {
        double r68378 = a;
        double r68379 = r68378 * r68378;
        double r68380 = b;
        double r68381 = r68380 * r68380;
        double r68382 = r68379 - r68381;
        double r68383 = r68382 / r68379;
        double r68384 = fabs(r68383);
        double r68385 = sqrt(r68384);
        return r68385;
}
↓
double f(double a, double b) {
        double r68386 = a;
        double r68387 = b;
        double r68388 = r68386 + r68387;
        double r68389 = r68388 / r68386;
        double r68390 = r68386 - r68387;
        double r68391 = r68390 / r68386;
        double r68392 = r68389 * r68391;
        double r68393 = exp(r68392);
        double r68394 = log(r68393);
        double r68395 = fabs(r68394);
        double r68396 = sqrt(r68395);
        return r68396;
}

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 14.2

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

  3. Applied difference-of-squares14.2

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

  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

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020020 +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)))))