Average Error: 0.0 → 0.0
Time: 11.5s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}\]
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}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r686224 = 1.0;
        double r686225 = 2.0;
        double r686226 = t;
        double r686227 = r686225 / r686226;
        double r686228 = r686224 / r686226;
        double r686229 = r686224 + r686228;
        double r686230 = r686227 / r686229;
        double r686231 = r686225 - r686230;
        double r686232 = r686231 * r686231;
        double r686233 = r686225 + r686232;
        double r686234 = r686224 / r686233;
        double r686235 = r686224 - r686234;
        return r686235;
}
↓
double f(double t) {
        double r686236 = 1.0;
        double r686237 = 2.0;
        double r686238 = t;
        double r686239 = r686236 + r686238;
        double r686240 = r686237 / r686239;
        double r686241 = r686237 - r686240;
        double r686242 = fma(r686241, r686241, r686237);
        double r686243 = r686236 / r686242;
        double r686244 = r686236 - r686243;
        return r686244;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019138 +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)))))))))