Average Error: 14.5 → 0.0
Time: 2.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{a - b}{a}\right)\right)}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{a - b}{a}\right)\right)}{a}\right|}
double f(double a, double b) {
        double r85065 = a;
        double r85066 = r85065 * r85065;
        double r85067 = b;
        double r85068 = r85067 * r85067;
        double r85069 = r85066 - r85068;
        double r85070 = r85069 / r85066;
        double r85071 = fabs(r85070);
        double r85072 = sqrt(r85071);
        return r85072;
}
↓
double f(double a, double b) {
        double r85073 = a;
        double r85074 = b;
        double r85075 = r85073 + r85074;
        double r85076 = r85073 - r85074;
        double r85077 = r85076 / r85073;
        double r85078 = expm1(r85077);
        double r85079 = log1p(r85078);
        double r85080 = r85075 * r85079;
        double r85081 = r85080 / r85073;
        double r85082 = fabs(r85081);
        double r85083 = sqrt(r85082);
        return r85083;
}

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 log1p-expm1-u0.0

    \[\leadsto \sqrt{\left|\frac{\left(a + b\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{a - b}{a}\right)\right)}}{a}\right|}\]

  10. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{\left(a + b\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{a - b}{a}\right)\right)}{a}\right|}\]

Reproduce

herbie shell --seed 2020065 +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)))))