Average Error: 14.4 → 0.0
Time: 13.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r63560 = a;
        double r63561 = r63560 * r63560;
        double r63562 = b;
        double r63563 = r63562 * r63562;
        double r63564 = r63561 - r63563;
        double r63565 = r63564 / r63561;
        double r63566 = fabs(r63565);
        double r63567 = sqrt(r63566);
        return r63567;
}
↓
double f(double a, double b) {
        double r63568 = a;
        double r63569 = b;
        double r63570 = r63568 + r63569;
        double r63571 = r63570 / r63568;
        double r63572 = r63568 - r63569;
        double r63573 = r63572 / r63568;
        double r63574 = r63571 * r63573;
        double r63575 = fabs(r63574);
        double r63576 = sqrt(r63575);
        return r63576;
}

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. Using strategy rm

  3. Applied difference-of-squares14.4

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019212 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))