Average Error: 14.5 → 0.0
Time: 2.8s
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 r78313 = a;
        double r78314 = r78313 * r78313;
        double r78315 = b;
        double r78316 = r78315 * r78315;
        double r78317 = r78314 - r78316;
        double r78318 = r78317 / r78314;
        double r78319 = fabs(r78318);
        double r78320 = sqrt(r78319);
        return r78320;
}
↓
double f(double a, double b) {
        double r78321 = a;
        double r78322 = b;
        double r78323 = r78321 + r78322;
        double r78324 = r78323 / r78321;
        double r78325 = r78321 - r78322;
        double r78326 = r78325 / r78321;
        double r78327 = r78324 * r78326;
        double r78328 = fabs(r78327);
        double r78329 = sqrt(r78328);
        return r78329;
}

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 14.5

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

  3. Applied difference-of-squares14.5

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

  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 2020065 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))