Average Error: 14.7 → 0.0
Time: 22.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a \cdot \frac{a}{b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a \cdot \frac{a}{b}}\right|}
double f(double a, double b) {
        double r66380 = a;
        double r66381 = r66380 * r66380;
        double r66382 = b;
        double r66383 = r66382 * r66382;
        double r66384 = r66381 - r66383;
        double r66385 = r66384 / r66381;
        double r66386 = fabs(r66385);
        double r66387 = sqrt(r66386);
        return r66387;
}
↓
double f(double a, double b) {
        double r66388 = 1.0;
        double r66389 = b;
        double r66390 = a;
        double r66391 = r66390 / r66389;
        double r66392 = r66390 * r66391;
        double r66393 = r66389 / r66392;
        double r66394 = r66388 - r66393;
        double r66395 = fabs(r66394);
        double r66396 = sqrt(r66395);
        return r66396;
}

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 14.7

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

  2. Simplified14.7

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

  4. Applied associate-/l*14.6

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))