Average Error: 0.1 → 0.1
Time: 3.7s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 \cdot 1 - \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}{\left(2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(1 - \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}\]
\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 \cdot 1 - \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(\frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}{\left(2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right) \cdot \left(1 - \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}\right)}
double f(double t) {
        double r77707 = 1.0;
        double r77708 = 2.0;
        double r77709 = t;
        double r77710 = r77708 * r77709;
        double r77711 = r77707 + r77709;
        double r77712 = r77710 / r77711;
        double r77713 = r77712 * r77712;
        double r77714 = r77707 + r77713;
        double r77715 = r77708 + r77713;
        double r77716 = r77714 / r77715;
        return r77716;
}
↓
double f(double t) {
        double r77717 = 1.0;
        double r77718 = r77717 * r77717;
        double r77719 = 2.0;
        double r77720 = t;
        double r77721 = r77719 * r77720;
        double r77722 = r77717 + r77720;
        double r77723 = r77721 / r77722;
        double r77724 = r77723 * r77723;
        double r77725 = r77724 * r77724;
        double r77726 = r77718 - r77725;
        double r77727 = r77719 + r77724;
        double r77728 = r77717 - r77724;
        double r77729 = r77727 * r77728;
        double r77730 = r77726 / r77729;
        return r77730;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip-+0.1

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

  4. Applied associate-/l/0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(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))))))