Average Error: 0.0 → 0.0
Time: 6.3s
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 r525308 = 1.0;
        double r525309 = 2.0;
        double r525310 = t;
        double r525311 = r525309 / r525310;
        double r525312 = r525308 / r525310;
        double r525313 = r525308 + r525312;
        double r525314 = r525311 / r525313;
        double r525315 = r525309 - r525314;
        double r525316 = r525315 * r525315;
        double r525317 = r525309 + r525316;
        double r525318 = r525308 / r525317;
        double r525319 = r525308 - r525318;
        return r525319;
}
↓
double f(double t) {
        double r525320 = 1.0;
        double r525321 = 2.0;
        double r525322 = t;
        double r525323 = r525320 + r525322;
        double r525324 = r525321 / r525323;
        double r525325 = r525321 - r525324;
        double r525326 = fma(r525325, r525325, r525321);
        double r525327 = r525320 / r525326;
        double r525328 = r525320 - r525327;
        return r525328;
}

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 2019154 +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)))))))))