Average Error: 14.7 → 0.0
Time: 3.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|e^{\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)} - 1\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|e^{\mathsf{log1p}\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)} - 1\right|}
double f(double a, double b) {
        double r69014 = a;
        double r69015 = r69014 * r69014;
        double r69016 = b;
        double r69017 = r69016 * r69016;
        double r69018 = r69015 - r69017;
        double r69019 = r69018 / r69015;
        double r69020 = fabs(r69019);
        double r69021 = sqrt(r69020);
        return r69021;
}
↓
double f(double a, double b) {
        double r69022 = a;
        double r69023 = b;
        double r69024 = r69022 + r69023;
        double r69025 = r69024 / r69022;
        double r69026 = r69022 - r69023;
        double r69027 = r69026 / r69022;
        double r69028 = r69025 * r69027;
        double r69029 = log1p(r69028);
        double r69030 = exp(r69029);
        double r69031 = 1.0;
        double r69032 = r69030 - r69031;
        double r69033 = fabs(r69032);
        double r69034 = sqrt(r69033);
        return r69034;
}

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 14.7

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

  3. Applied difference-of-squares14.7

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

  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 expm1-log1p-u0.0

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

  8. Applied expm1-udef0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 +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)))))