Average Error: 14.4 → 0.0
Time: 4.4s
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 r79807 = a;
        double r79808 = r79807 * r79807;
        double r79809 = b;
        double r79810 = r79809 * r79809;
        double r79811 = r79808 - r79810;
        double r79812 = r79811 / r79808;
        double r79813 = fabs(r79812);
        double r79814 = sqrt(r79813);
        return r79814;
}
↓
double f(double a, double b) {
        double r79815 = 1.0;
        double r79816 = b;
        double r79817 = a;
        double r79818 = r79816 / r79817;
        double r79819 = r79818 * r79818;
        double r79820 = r79815 - r79819;
        double r79821 = fabs(r79820);
        double r79822 = sqrt(r79821);
        return r79822;
}

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 14.4

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

  3. Applied difference-of-squares14.4

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

  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 div-sub0.0

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

  7. Simplified0.0

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

  8. Taylor expanded around 0 14.4

    \[\leadsto \sqrt{\left|\color{blue}{1 - \frac{{b}^{2}}{{a}^{2}}}\right|}\]

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019318 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))