Average Error: 14.3 → 0.0
Time: 10.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|e^{\mathsf{log1p}\left(\frac{b}{a} \cdot \left(-\frac{b}{a}\right)\right)}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|e^{\mathsf{log1p}\left(\frac{b}{a} \cdot \left(-\frac{b}{a}\right)\right)}\right|}
double f(double a, double b) {
        double r57044 = a;
        double r57045 = r57044 * r57044;
        double r57046 = b;
        double r57047 = r57046 * r57046;
        double r57048 = r57045 - r57047;
        double r57049 = r57048 / r57045;
        double r57050 = fabs(r57049);
        double r57051 = sqrt(r57050);
        return r57051;
}
↓
double f(double a, double b) {
        double r57052 = b;
        double r57053 = a;
        double r57054 = r57052 / r57053;
        double r57055 = -r57054;
        double r57056 = r57054 * r57055;
        double r57057 = log1p(r57056);
        double r57058 = exp(r57057);
        double r57059 = fabs(r57058);
        double r57060 = sqrt(r57059);
        return r57060;
}

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. Simplified14.3

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

  4. Applied add-exp-log14.3

    \[\leadsto \sqrt{\left|\color{blue}{e^{\log \left(1 - \frac{b \cdot b}{a \cdot a}\right)}}\right|}\]

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019209 +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)))))