Average Error: 29.5 → 7.9
Time: 29.0s
Precision: 64
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
double f(double a, double b) {
        double r8433150 = a;
        double r8433151 = r8433150 * r8433150;
        double r8433152 = b;
        double r8433153 = r8433152 * r8433152;
        double r8433154 = r8433151 - r8433153;
        double r8433155 = r8433154 / r8433151;
        double r8433156 = fabs(r8433155);
        double r8433157 = sqrt(r8433156);
        return r8433157;
}
↓
double f(double a, double b) {
        double r8433158 = 1.0;
        double r8433159 = b;
        double r8433160 = a;
        double r8433161 = r8433159 / r8433160;
        double r8433162 = r8433161 * r8433161;
        double r8433163 = r8433158 - r8433162;
        double r8433164 = fabs(r8433163);
        double r8433165 = sqrt(r8433164);
        return r8433165;
}


\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 29.5

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

  2. Simplified7.9

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

  3. Final simplification7.9

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

Reproduce

herbie shell --seed 2019102 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))