Average Error: 14.5 → 0.0
Time: 3.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{\frac{a}{\frac{a - b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{\frac{a}{\frac{a - b}{a}}}\right|}
double f(double a, double b) {
        double r72103 = a;
        double r72104 = r72103 * r72103;
        double r72105 = b;
        double r72106 = r72105 * r72105;
        double r72107 = r72104 - r72106;
        double r72108 = r72107 / r72104;
        double r72109 = fabs(r72108);
        double r72110 = sqrt(r72109);
        return r72110;
}
↓
double f(double a, double b) {
        double r72111 = a;
        double r72112 = b;
        double r72113 = r72111 + r72112;
        double r72114 = r72111 - r72112;
        double r72115 = r72114 / r72111;
        double r72116 = r72111 / r72115;
        double r72117 = r72113 / r72116;
        double r72118 = fabs(r72117);
        double r72119 = sqrt(r72118);
        return r72119;
}

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 14.5

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

  3. Applied difference-of-squares14.5

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

  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. Using strategy rm

  9. Applied associate-/l*0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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)))))