Average Error: 14.6 → 0.0
Time: 3.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\sqrt[3]{{\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)}^{3}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\sqrt[3]{{\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)}^{3}}\right|}
double f(double a, double b) {
        double r62254 = a;
        double r62255 = r62254 * r62254;
        double r62256 = b;
        double r62257 = r62256 * r62256;
        double r62258 = r62255 - r62257;
        double r62259 = r62258 / r62255;
        double r62260 = fabs(r62259);
        double r62261 = sqrt(r62260);
        return r62261;
}
↓
double f(double a, double b) {
        double r62262 = a;
        double r62263 = b;
        double r62264 = r62262 + r62263;
        double r62265 = r62264 / r62262;
        double r62266 = r62262 - r62263;
        double r62267 = r62266 / r62262;
        double r62268 = r62265 * r62267;
        double r62269 = 3.0;
        double r62270 = pow(r62268, r62269);
        double r62271 = cbrt(r62270);
        double r62272 = fabs(r62271);
        double r62273 = sqrt(r62272);
        return r62273;
}

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 14.6

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

  3. Applied difference-of-squares14.6

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

  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-cbrt-cube27.8

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{\color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}}}\right|}\]

  7. Applied add-cbrt-cube27.2

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \frac{\color{blue}{\sqrt[3]{\left(\left(a - b\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right)}}}{\sqrt[3]{\left(a \cdot a\right) \cdot a}}\right|}\]

  8. Applied cbrt-undiv27.2

    \[\leadsto \sqrt{\left|\frac{a + b}{a} \cdot \color{blue}{\sqrt[3]{\frac{\left(\left(a - b\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right)}{\left(a \cdot a\right) \cdot a}}}\right|}\]

  9. Applied add-cbrt-cube27.8

    \[\leadsto \sqrt{\left|\frac{a + b}{\color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}}} \cdot \sqrt[3]{\frac{\left(\left(a - b\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right)}{\left(a \cdot a\right) \cdot a}}\right|}\]

  10. Applied add-cbrt-cube27.2

    \[\leadsto \sqrt{\left|\frac{\color{blue}{\sqrt[3]{\left(\left(a + b\right) \cdot \left(a + b\right)\right) \cdot \left(a + b\right)}}}{\sqrt[3]{\left(a \cdot a\right) \cdot a}} \cdot \sqrt[3]{\frac{\left(\left(a - b\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right)}{\left(a \cdot a\right) \cdot a}}\right|}\]

  11. Applied cbrt-undiv27.2

    \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{\frac{\left(\left(a + b\right) \cdot \left(a + b\right)\right) \cdot \left(a + b\right)}{\left(a \cdot a\right) \cdot a}}} \cdot \sqrt[3]{\frac{\left(\left(a - b\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right)}{\left(a \cdot a\right) \cdot a}}\right|}\]

  12. Applied cbrt-unprod27.2

    \[\leadsto \sqrt{\left|\color{blue}{\sqrt[3]{\frac{\left(\left(a + b\right) \cdot \left(a + b\right)\right) \cdot \left(a + b\right)}{\left(a \cdot a\right) \cdot a} \cdot \frac{\left(\left(a - b\right) \cdot \left(a - b\right)\right) \cdot \left(a - b\right)}{\left(a \cdot a\right) \cdot a}}}\right|}\]

  13. Simplified0.0

    \[\leadsto \sqrt{\left|\sqrt[3]{\color{blue}{{\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)}^{3}}}\right|}\]

  14. Final simplification0.0

    \[\leadsto \sqrt{\left|\sqrt[3]{{\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)}^{3}}\right|}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))