Average Error: 13.8 → 0.0
Time: 2.4s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r66058 = a;
        double r66059 = r66058 * r66058;
        double r66060 = b;
        double r66061 = r66060 * r66060;
        double r66062 = r66059 - r66061;
        double r66063 = r66062 / r66059;
        double r66064 = fabs(r66063);
        double r66065 = sqrt(r66064);
        return r66065;
}
↓
double f(double a, double b) {
        double r66066 = 1.0;
        double r66067 = b;
        double r66068 = a;
        double r66069 = r66067 / r66068;
        double r66070 = r66066 + r66069;
        double r66071 = r66068 - r66067;
        double r66072 = r66071 / r66068;
        double r66073 = r66070 * r66072;
        double r66074 = fabs(r66073);
        double r66075 = sqrt(r66074);
        return r66075;
}

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 13.8

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

  3. Applied difference-of-squares13.8

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

  4. Applied times-frac0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020003 +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)))))