Average Error: 14.5 → 0.0
Time: 3.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\log \left(e^{\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\log \left(e^{\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}}\right)\right|}
double f(double a, double b) {
        double r67041 = a;
        double r67042 = r67041 * r67041;
        double r67043 = b;
        double r67044 = r67043 * r67043;
        double r67045 = r67042 - r67044;
        double r67046 = r67045 / r67042;
        double r67047 = fabs(r67046);
        double r67048 = sqrt(r67047);
        return r67048;
}
↓
double f(double a, double b) {
        double r67049 = a;
        double r67050 = b;
        double r67051 = r67049 + r67050;
        double r67052 = r67049 - r67050;
        double r67053 = r67052 / r67049;
        double r67054 = r67051 * r67053;
        double r67055 = r67054 / r67049;
        double r67056 = exp(r67055);
        double r67057 = log(r67056);
        double r67058 = fabs(r67057);
        double r67059 = sqrt(r67058);
        return r67059;
}

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 14.5

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

  3. Applied difference-of-squares14.5

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

  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 associate-*r/0.0

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

  7. Simplified0.0

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

  9. Applied add-log-exp0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020064 +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)))))