Average Error: 0.0 → 0.1
Time: 21.5s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}\]
\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 r62181 = 1.0;
        double r62182 = 2.0;
        double r62183 = t;
        double r62184 = r62182 * r62183;
        double r62185 = r62181 + r62183;
        double r62186 = r62184 / r62185;
        double r62187 = r62186 * r62186;
        double r62188 = r62181 + r62187;
        double r62189 = r62182 + r62187;
        double r62190 = r62188 / r62189;
        return r62190;
}
↓
double f(double t) {
        double r62191 = 1.0;
        double r62192 = 2.0;
        double r62193 = t;
        double r62194 = r62192 * r62193;
        double r62195 = r62191 + r62193;
        double r62196 = r62194 / r62195;
        double r62197 = r62194 * r62196;
        double r62198 = r62197 / r62195;
        double r62199 = r62191 + r62198;
        double r62200 = r62192 + r62198;
        double r62201 = r62199 / r62200;
        return r62201;
}

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))))))