Average Error: 30.3 → 8.1
Time: 9.4s
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 r38454 = a;
        double r38455 = r38454 * r38454;
        double r38456 = b;
        double r38457 = r38456 * r38456;
        double r38458 = r38455 - r38457;
        double r38459 = r38458 / r38455;
        double r38460 = fabs(r38459);
        double r38461 = sqrt(r38460);
        return r38461;
}
↓
double f(double a, double b) {
        double r38462 = 1.0;
        double r38463 = b;
        double r38464 = a;
        double r38465 = r38463 / r38464;
        double r38466 = r38465 * r38465;
        double r38467 = r38462 - r38466;
        double r38468 = fabs(r38467);
        double r38469 = sqrt(r38468);
        return r38469;
}

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 30.3

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

  2. Simplified14.7

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

  4. Applied div-sub14.7

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

  5. Simplified14.7

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

  6. Simplified8.1

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

  7. Final simplification8.1

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

Reproduce

herbie shell --seed 2019179 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))