Average Error: 15.0 → 0.0
Time: 11.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\log \left(e^{\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}}\right)\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\log \left(e^{\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}}\right)
double f(double a, double b) {
        double r54495 = a;
        double r54496 = r54495 * r54495;
        double r54497 = b;
        double r54498 = r54497 * r54497;
        double r54499 = r54496 - r54498;
        double r54500 = r54499 / r54496;
        double r54501 = fabs(r54500);
        double r54502 = sqrt(r54501);
        return r54502;
}
↓
double f(double a, double b) {
        double r54503 = a;
        double r54504 = b;
        double r54505 = r54503 + r54504;
        double r54506 = r54505 / r54503;
        double r54507 = r54503 - r54504;
        double r54508 = r54507 / r54503;
        double r54509 = r54506 * r54508;
        double r54510 = fabs(r54509);
        double r54511 = sqrt(r54510);
        double r54512 = exp(r54511);
        double r54513 = log(r54512);
        return r54513;
}

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 15.0

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

  3. Applied difference-of-squares15.0

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

  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 add-log-exp0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))