Average Error: 0.0 → 0.1
Time: 23.8s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}\]
\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
↓
\frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}
double f(double t) {
        double r62486 = 1.0;
        double r62487 = 2.0;
        double r62488 = t;
        double r62489 = r62487 * r62488;
        double r62490 = r62486 + r62488;
        double r62491 = r62489 / r62490;
        double r62492 = r62491 * r62491;
        double r62493 = r62486 + r62492;
        double r62494 = r62487 + r62492;
        double r62495 = r62493 / r62494;
        return r62495;
}
↓
double f(double t) {
        double r62496 = 1.0;
        double r62497 = 2.0;
        double r62498 = t;
        double r62499 = r62497 * r62498;
        double r62500 = r62496 + r62498;
        double r62501 = r62499 / r62500;
        double r62502 = r62499 * r62501;
        double r62503 = r62502 / r62500;
        double r62504 = r62496 + r62503;
        double r62505 = r62497 + r62503;
        double r62506 = r62504 / r62505;
        return r62506;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
  2. Using strategy rm

  3. Applied associate-*r/0.1

    \[\leadsto \frac{1 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  4. Simplified0.1

    \[\leadsto \frac{1 + \frac{\color{blue}{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
  5. Using strategy rm

  6. Applied associate-*r/0.1

    \[\leadsto \frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \color{blue}{\frac{\frac{2 \cdot t}{1 + t} \cdot \left(2 \cdot t\right)}{1 + t}}}\]

  7. Simplified0.1

    \[\leadsto \frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \frac{\color{blue}{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}}{1 + t}}\]

  8. Final simplification0.1

    \[\leadsto \frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (t)
  :name "Kahan p13 Example 1"
  :precision binary64
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))