Average Error: 14.0 → 0.0
Time: 22.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \left|\frac{b}{a}\right| \cdot \left|\frac{b}{a}\right|\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \left|\frac{b}{a}\right| \cdot \left|\frac{b}{a}\right|\right|}
double f(double a, double b) {
        double r39311 = a;
        double r39312 = r39311 * r39311;
        double r39313 = b;
        double r39314 = r39313 * r39313;
        double r39315 = r39312 - r39314;
        double r39316 = r39315 / r39312;
        double r39317 = fabs(r39316);
        double r39318 = sqrt(r39317);
        return r39318;
}
↓
double f(double a, double b) {
        double r39319 = 1.0;
        double r39320 = b;
        double r39321 = a;
        double r39322 = r39320 / r39321;
        double r39323 = fabs(r39322);
        double r39324 = r39323 * r39323;
        double r39325 = r39319 - r39324;
        double r39326 = fabs(r39325);
        double r39327 = sqrt(r39326);
        return r39327;
}

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 add-sqr-sqrt14.0

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

  5. Simplified14.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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)))))