Average Error: 29.5 → 7.4
Time: 1.0m
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r11688479 = a;
        double r11688480 = r11688479 * r11688479;
        double r11688481 = b;
        double r11688482 = r11688481 * r11688481;
        double r11688483 = r11688480 - r11688482;
        double r11688484 = r11688483 / r11688480;
        double r11688485 = fabs(r11688484);
        double r11688486 = sqrt(r11688485);
        return r11688486;
}
↓
double f(double a, double b) {
        double r11688487 = 1.0;
        double r11688488 = b;
        double r11688489 = a;
        double r11688490 = r11688488 / r11688489;
        double r11688491 = r11688490 * r11688490;
        double r11688492 = r11688487 - r11688491;
        double r11688493 = fabs(r11688492);
        double r11688494 = sqrt(r11688493);
        return r11688494;
}

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 29.5

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

  2. Simplified7.4

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

  3. Final simplification7.4

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

Reproduce

herbie shell --seed 2019125 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))