Average Error: 14.1 → 0.0
Time: 3.8s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \frac{1 - \frac{b}{a}}{a}\right)\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{expm1}\left(\mathsf{log1p}\left(\left(a + b\right) \cdot \frac{1 - \frac{b}{a}}{a}\right)\right)\right|}
double f(double a, double b) {
        double r105195 = a;
        double r105196 = r105195 * r105195;
        double r105197 = b;
        double r105198 = r105197 * r105197;
        double r105199 = r105196 - r105198;
        double r105200 = r105199 / r105196;
        double r105201 = fabs(r105200);
        double r105202 = sqrt(r105201);
        return r105202;
}
↓
double f(double a, double b) {
        double r105203 = a;
        double r105204 = b;
        double r105205 = r105203 + r105204;
        double r105206 = 1.0;
        double r105207 = r105204 / r105203;
        double r105208 = r105206 - r105207;
        double r105209 = r105208 / r105203;
        double r105210 = r105205 * r105209;
        double r105211 = log1p(r105210);
        double r105212 = expm1(r105211);
        double r105213 = fabs(r105212);
        double r105214 = sqrt(r105213);
        return r105214;
}

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 14.1

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

  3. Applied difference-of-squares14.1

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

  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 div-inv0.0

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

  9. Applied associate-*l*0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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)))))