Average Error: 0.1 → 0.1
Time: 8.5s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 1)_*}{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 2)_*}\]
\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{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 1)_*}{(\left(\frac{t \cdot 2}{1 + t}\right) \cdot \left(\frac{t \cdot 2}{1 + t}\right) + 2)_*}
double f(double t) {
        double r3183354 = 1.0;
        double r3183355 = 2.0;
        double r3183356 = t;
        double r3183357 = r3183355 * r3183356;
        double r3183358 = r3183354 + r3183356;
        double r3183359 = r3183357 / r3183358;
        double r3183360 = r3183359 * r3183359;
        double r3183361 = r3183354 + r3183360;
        double r3183362 = r3183355 + r3183360;
        double r3183363 = r3183361 / r3183362;
        return r3183363;
}
↓
double f(double t) {
        double r3183364 = t;
        double r3183365 = 2.0;
        double r3183366 = r3183364 * r3183365;
        double r3183367 = 1.0;
        double r3183368 = r3183367 + r3183364;
        double r3183369 = r3183366 / r3183368;
        double r3183370 = fma(r3183369, r3183369, r3183367);
        double r3183371 = fma(r3183369, r3183369, r3183365);
        double r3183372 = r3183370 / r3183371;
        return r3183372;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.1

    \[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 1"
  (/ (+ 1 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t)))) (+ 2 (* (/ (* 2 t) (+ 1 t)) (/ (* 2 t) (+ 1 t))))))