Average Error: 0.1 → 0.0
Time: 9.2s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)}{2 + \left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{1 + \left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)}{2 + \left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)}
double f(double t) {
        double r43793 = 1.0;
        double r43794 = 2.0;
        double r43795 = t;
        double r43796 = r43794 / r43795;
        double r43797 = r43793 / r43795;
        double r43798 = r43793 + r43797;
        double r43799 = r43796 / r43798;
        double r43800 = r43794 - r43799;
        double r43801 = r43800 * r43800;
        double r43802 = r43793 + r43801;
        double r43803 = r43794 + r43801;
        double r43804 = r43802 / r43803;
        return r43804;
}
↓
double f(double t) {
        double r43805 = 1.0;
        double r43806 = 2.0;
        double r43807 = t;
        double r43808 = r43807 * r43805;
        double r43809 = r43805 + r43808;
        double r43810 = r43806 / r43809;
        double r43811 = r43806 - r43810;
        double r43812 = r43811 * r43811;
        double r43813 = r43805 + r43812;
        double r43814 = r43806 + r43812;
        double r43815 = r43813 / r43814;
        return r43815;
}

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

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))