Average Error: 0.0 → 0.0
Time: 8.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(t, 1, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(t, 1, 1\right)}, 2\right)}\]
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(2 - \frac{2}{\mathsf{fma}\left(t, 1, 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(t, 1, 1\right)}, 2\right)}
double f(double t) {
        double r19952 = 1.0;
        double r19953 = 2.0;
        double r19954 = t;
        double r19955 = r19953 / r19954;
        double r19956 = r19952 / r19954;
        double r19957 = r19952 + r19956;
        double r19958 = r19955 / r19957;
        double r19959 = r19953 - r19958;
        double r19960 = r19959 * r19959;
        double r19961 = r19953 + r19960;
        double r19962 = r19952 / r19961;
        double r19963 = r19952 - r19962;
        return r19963;
}
↓
double f(double t) {
        double r19964 = 1.0;
        double r19965 = 2.0;
        double r19966 = t;
        double r19967 = fma(r19966, r19964, r19964);
        double r19968 = r19965 / r19967;
        double r19969 = r19965 - r19968;
        double r19970 = fma(r19969, r19969, r19965);
        double r19971 = r19964 / r19970;
        double r19972 = r19964 - r19971;
        return r19972;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))