Average Error: 0.0 → 0.1
Time: 23.6s
Precision: 64
\[\frac{1 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
↓
\[\frac{1 + \frac{\left(2 \cdot t\right) \cdot \frac{2 \cdot t}{1 + t}}{1 + t}}{2 + \frac{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}\]
\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{2 \cdot t}{1 + t} \cdot \frac{2 \cdot t}{1 + t}}
double f(double t) {
        double r59859 = 1.0;
        double r59860 = 2.0;
        double r59861 = t;
        double r59862 = r59860 * r59861;
        double r59863 = r59859 + r59861;
        double r59864 = r59862 / r59863;
        double r59865 = r59864 * r59864;
        double r59866 = r59859 + r59865;
        double r59867 = r59860 + r59865;
        double r59868 = r59866 / r59867;
        return r59868;
}
↓
double f(double t) {
        double r59869 = 1.0;
        double r59870 = 2.0;
        double r59871 = t;
        double r59872 = r59870 * r59871;
        double r59873 = r59869 + r59871;
        double r59874 = r59872 / r59873;
        double r59875 = r59872 * r59874;
        double r59876 = r59875 / r59873;
        double r59877 = r59869 + r59876;
        double r59878 = r59874 * r59874;
        double r59879 = r59870 + r59878;
        double r59880 = r59877 / r59879;
        return r59880;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 +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))))))