Average Error: 14.3 → 0.0
Time: 4.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\]
\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 r92028 = a;
        double r92029 = r92028 * r92028;
        double r92030 = b;
        double r92031 = r92030 * r92030;
        double r92032 = r92029 - r92031;
        double r92033 = r92032 / r92029;
        double r92034 = fabs(r92033);
        double r92035 = sqrt(r92034);
        return r92035;
}
↓
double f(double a, double b) {
        double r92036 = a;
        double r92037 = b;
        double r92038 = r92036 + r92037;
        double r92039 = r92038 / r92036;
        double r92040 = r92036 - r92037;
        double r92041 = r92040 / r92036;
        double r92042 = r92039 * r92041;
        double r92043 = fabs(r92042);
        double r92044 = sqrt(r92043);
        return r92044;
}

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 14.3

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

  3. Applied difference-of-squares14.3

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

  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 2020046 +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)))))