Average Error: 14.0 → 0.0
Time: 10.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{\frac{b}{a} \cdot \frac{b}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{\frac{b}{a} \cdot \frac{b}{a}}\right)\right|}
double f(double a, double b) {
        double r53175 = a;
        double r53176 = r53175 * r53175;
        double r53177 = b;
        double r53178 = r53177 * r53177;
        double r53179 = r53176 - r53178;
        double r53180 = r53179 / r53176;
        double r53181 = fabs(r53180);
        double r53182 = sqrt(r53181);
        return r53182;
}
↓
double f(double a, double b) {
        double r53183 = 1.0;
        double r53184 = b;
        double r53185 = a;
        double r53186 = r53184 / r53185;
        double r53187 = r53186 * r53186;
        double r53188 = exp(r53187);
        double r53189 = log(r53188);
        double r53190 = r53183 - r53189;
        double r53191 = fabs(r53190);
        double r53192 = sqrt(r53191);
        return r53192;
}

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 14.0

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

  2. Simplified14.0

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b \cdot b}{a \cdot a}\right|}}\]
  3. Using strategy rm

  4. Applied times-frac0.0

    \[\leadsto \sqrt{\left|1 - \color{blue}{\frac{b}{a} \cdot \frac{b}{a}}\right|}\]
  5. Using strategy rm

  6. Applied add-log-exp0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +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)))))