Average Error: 15.0 → 0.0
Time: 13.5s
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 r44470 = a;
        double r44471 = r44470 * r44470;
        double r44472 = b;
        double r44473 = r44472 * r44472;
        double r44474 = r44471 - r44473;
        double r44475 = r44474 / r44471;
        double r44476 = fabs(r44475);
        double r44477 = sqrt(r44476);
        return r44477;
}
↓
double f(double a, double b) {
        double r44478 = 1.0;
        double r44479 = b;
        double r44480 = a;
        double r44481 = r44479 / r44480;
        double r44482 = fabs(r44481);
        double r44483 = r44482 * r44482;
        double r44484 = r44478 - r44483;
        double r44485 = fabs(r44484);
        double r44486 = sqrt(r44485);
        return r44486;
}

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 15.0

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

  2. Simplified15.0

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

  4. Applied add-sqr-sqrt15.0

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

  5. Simplified15.0

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

  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 2020047 +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)))))