Average Error: 15.0 → 0.0
Time: 3.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt[3]{{\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)}^{3}}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt[3]{{\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)}^{3}}
double f(double a, double b) {
        double r72532 = a;
        double r72533 = r72532 * r72532;
        double r72534 = b;
        double r72535 = r72534 * r72534;
        double r72536 = r72533 - r72535;
        double r72537 = r72536 / r72533;
        double r72538 = fabs(r72537);
        double r72539 = sqrt(r72538);
        return r72539;
}
↓
double f(double a, double b) {
        double r72540 = a;
        double r72541 = b;
        double r72542 = r72540 + r72541;
        double r72543 = r72542 / r72540;
        double r72544 = r72540 - r72541;
        double r72545 = r72544 / r72540;
        double r72546 = r72543 * r72545;
        double r72547 = fabs(r72546);
        double r72548 = sqrt(r72547);
        double r72549 = 3.0;
        double r72550 = pow(r72548, r72549);
        double r72551 = cbrt(r72550);
        return r72551;
}

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-cbrt-cube0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

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)))))