Average Error: 0.0 → 0.0
Time: 1.2m
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
1 - \frac{1}{\mathsf{fma}\left(\left(2 - \frac{2}{1 + t}\right), \left(2 - \frac{2}{1 + t}\right), 2\right)}
double f(double t) {
        double r3346955 = 1.0;
        double r3346956 = 2.0;
        double r3346957 = t;
        double r3346958 = r3346956 / r3346957;
        double r3346959 = r3346955 / r3346957;
        double r3346960 = r3346955 + r3346959;
        double r3346961 = r3346958 / r3346960;
        double r3346962 = r3346956 - r3346961;
        double r3346963 = r3346962 * r3346962;
        double r3346964 = r3346956 + r3346963;
        double r3346965 = r3346955 / r3346964;
        double r3346966 = r3346955 - r3346965;
        return r3346966;
}
↓
double f(double t) {
        double r3346967 = 1.0;
        double r3346968 = 2.0;
        double r3346969 = t;
        double r3346970 = r3346967 + r3346969;
        double r3346971 = r3346968 / r3346970;
        double r3346972 = r3346968 - r3346971;
        double r3346973 = fma(r3346972, r3346972, r3346968);
        double r3346974 = r3346967 / r3346973;
        double r3346975 = r3346967 - r3346974;
        return r3346975;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))