Average Error: 0.0 → 0.0
Time: 13.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}\]
double f(double t) {
        double r1654375 = 1.0;
        double r1654376 = 2.0;
        double r1654377 = t;
        double r1654378 = r1654376 / r1654377;
        double r1654379 = r1654375 / r1654377;
        double r1654380 = r1654375 + r1654379;
        double r1654381 = r1654378 / r1654380;
        double r1654382 = r1654376 - r1654381;
        double r1654383 = r1654382 * r1654382;
        double r1654384 = r1654376 + r1654383;
        double r1654385 = r1654375 / r1654384;
        double r1654386 = r1654375 - r1654385;
        return r1654386;
}
↓
double f(double t) {
        double r1654387 = 1.0;
        double r1654388 = 2.0;
        double r1654389 = t;
        double r1654390 = r1654387 + r1654389;
        double r1654391 = r1654388 / r1654390;
        double r1654392 = r1654388 - r1654391;
        double r1654393 = r1654392 * r1654392;
        double r1654394 = r1654388 + r1654393;
        double r1654395 = r1654387 / r1654394;
        double r1654396 = r1654387 - r1654395;
        return r1654396;
}


1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))