Average Error: 14.1 → 0.0
Time: 4.8s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r106113 = a;
        double r106114 = r106113 * r106113;
        double r106115 = b;
        double r106116 = r106115 * r106115;
        double r106117 = r106114 - r106116;
        double r106118 = r106117 / r106114;
        double r106119 = fabs(r106118);
        double r106120 = sqrt(r106119);
        return r106120;
}
↓
double f(double a, double b) {
        double r106121 = 1.0;
        double r106122 = a;
        double r106123 = b;
        double r106124 = r106122 + r106123;
        double r106125 = r106122 / r106124;
        double r106126 = r106121 / r106125;
        double r106127 = r106122 - r106123;
        double r106128 = r106127 / r106122;
        double r106129 = r106126 * r106128;
        double r106130 = fabs(r106129);
        double r106131 = sqrt(r106130);
        return r106131;
}

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 14.1

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

  3. Applied difference-of-squares14.1

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

  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 clear-num0.0

    \[\leadsto \sqrt{\left|\color{blue}{\frac{1}{\frac{a}{a + b}}} \cdot \frac{a - b}{a}\right|}\]

  7. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{1}{\frac{a}{a + b}} \cdot \frac{a - b}{a}\right|}\]

Reproduce

herbie shell --seed 2020035 +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)))))