Average Error: 14.8 → 0.0
Time: 3.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}
double f(double a, double b) {
        double r75103 = a;
        double r75104 = r75103 * r75103;
        double r75105 = b;
        double r75106 = r75105 * r75105;
        double r75107 = r75104 - r75106;
        double r75108 = r75107 / r75104;
        double r75109 = fabs(r75108);
        double r75110 = sqrt(r75109);
        return r75110;
}
↓
double f(double a, double b) {
        double r75111 = a;
        double r75112 = b;
        double r75113 = r75111 + r75112;
        double r75114 = r75111 - r75112;
        double r75115 = r75114 / r75111;
        double r75116 = r75113 * r75115;
        double r75117 = r75116 / r75111;
        double r75118 = fabs(r75117);
        double r75119 = sqrt(r75118);
        return r75119;
}

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 14.8

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares14.8

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

  4. Applied times-frac0.0

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

  6. Applied associate-*r/0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}\]

Reproduce

herbie shell --seed 2020059 +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)))))