Average Error: 14.3 → 0.0
Time: 12.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\right|}
double f(double a, double b) {
        double r58173 = a;
        double r58174 = r58173 * r58173;
        double r58175 = b;
        double r58176 = r58175 * r58175;
        double r58177 = r58174 - r58176;
        double r58178 = r58177 / r58174;
        double r58179 = fabs(r58178);
        double r58180 = sqrt(r58179);
        return r58180;
}
↓
double f(double a, double b) {
        double r58181 = 1.0;
        double r58182 = b;
        double r58183 = a;
        double r58184 = r58182 / r58183;
        double r58185 = r58184 * r58184;
        double r58186 = r58181 - r58185;
        double r58187 = log1p(r58186);
        double r58188 = expm1(r58187);
        double r58189 = fabs(r58188);
        double r58190 = sqrt(r58189);
        return r58190;
}

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 14.3

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

  3. Applied clear-num14.3

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

  4. Simplified0.0

    \[\leadsto \sqrt{\left|\frac{1}{\color{blue}{\frac{a}{a - b \cdot \frac{b}{a}}}}\right|}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.0

    \[\leadsto \sqrt{\left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{a}{a - b \cdot \frac{b}{a}}}\right)\right)}\right|}\]

  7. Simplified0.0

    \[\leadsto \sqrt{\left|\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)}\right)\right|}\]

  8. Final simplification0.0

    \[\leadsto \sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{b}{a} \cdot \frac{b}{a}\right)\right)\right|}\]

Reproduce

herbie shell --seed 2019350 +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)))))