Average Error: 31.3 → 8.2
Time: 14.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}
double f(double a, double b) {
        double r3408083 = a;
        double r3408084 = r3408083 * r3408083;
        double r3408085 = b;
        double r3408086 = r3408085 * r3408085;
        double r3408087 = r3408084 - r3408086;
        double r3408088 = r3408087 / r3408084;
        double r3408089 = fabs(r3408088);
        double r3408090 = sqrt(r3408089);
        return r3408090;
}
↓
double f(double a, double b) {
        double r3408091 = 1.0;
        double r3408092 = b;
        double r3408093 = a;
        double r3408094 = r3408092 / r3408093;
        double r3408095 = r3408094 * r3408094;
        double r3408096 = r3408091 - r3408095;
        double r3408097 = fabs(r3408096);
        double r3408098 = sqrt(r3408097);
        return r3408098;
}

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 31.3

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified8.2

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]

  3. Final simplification8.2

    \[\leadsto \sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))