Average Error: 14.6 → 0.0
Time: 3.8s
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 r77712 = a;
        double r77713 = r77712 * r77712;
        double r77714 = b;
        double r77715 = r77714 * r77714;
        double r77716 = r77713 - r77715;
        double r77717 = r77716 / r77713;
        double r77718 = fabs(r77717);
        double r77719 = sqrt(r77718);
        return r77719;
}
↓
double f(double a, double b) {
        double r77720 = a;
        double r77721 = b;
        double r77722 = r77720 + r77721;
        double r77723 = r77720 - r77721;
        double r77724 = r77723 / r77720;
        double r77725 = r77720 / r77724;
        double r77726 = r77722 / r77725;
        double r77727 = fabs(r77726);
        double r77728 = sqrt(r77727);
        return r77728;
}

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 14.6

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

  3. Applied difference-of-squares14.6

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

  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 2020036 +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)))))