Average Error: 0.0 → 0.0
Time: 8.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}\]
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(1, 1, t \cdot 1\right)}, 2 - \frac{2}{\mathsf{fma}\left(1, 1, t \cdot 1\right)}, 2\right)}
double f(double t) {
        double r24245 = 1.0;
        double r24246 = 2.0;
        double r24247 = t;
        double r24248 = r24246 / r24247;
        double r24249 = r24245 / r24247;
        double r24250 = r24245 + r24249;
        double r24251 = r24248 / r24250;
        double r24252 = r24246 - r24251;
        double r24253 = r24252 * r24252;
        double r24254 = r24246 + r24253;
        double r24255 = r24245 / r24254;
        double r24256 = r24245 - r24255;
        return r24256;
}
↓
double f(double t) {
        double r24257 = 1.0;
        double r24258 = 2.0;
        double r24259 = 1.0;
        double r24260 = t;
        double r24261 = r24260 * r24257;
        double r24262 = fma(r24259, r24257, r24261);
        double r24263 = r24258 / r24262;
        double r24264 = r24258 - r24263;
        double r24265 = fma(r24264, r24264, r24258);
        double r24266 = r24257 / r24265;
        double r24267 = r24257 - r24266;
        return r24267;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 +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)))))))))