Average Error: 14.8 → 0.0
Time: 25.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r27212 = a;
        double r27213 = r27212 * r27212;
        double r27214 = b;
        double r27215 = r27214 * r27214;
        double r27216 = r27213 - r27215;
        double r27217 = r27216 / r27213;
        double r27218 = fabs(r27217);
        double r27219 = sqrt(r27218);
        return r27219;
}
↓
double f(double a, double b) {
        double r27220 = 1.0;
        double r27221 = b;
        double r27222 = a;
        double r27223 = r27221 / r27222;
        double r27224 = r27223 * r27223;
        double r27225 = r27220 - r27224;
        double r27226 = fabs(r27225);
        double r27227 = sqrt(r27226);
        return r27227;
}

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 14.8

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

  2. Simplified14.8

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +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)))))