Average Error: 0.0 → 0.0
Time: 5.8s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}\]
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 r444091 = 1.0;
        double r444092 = 2.0;
        double r444093 = t;
        double r444094 = r444092 / r444093;
        double r444095 = r444091 / r444093;
        double r444096 = r444091 + r444095;
        double r444097 = r444094 / r444096;
        double r444098 = r444092 - r444097;
        double r444099 = r444098 * r444098;
        double r444100 = r444092 + r444099;
        double r444101 = r444091 / r444100;
        double r444102 = r444091 - r444101;
        return r444102;
}
↓
double f(double t) {
        double r444103 = 1.0;
        double r444104 = 2.0;
        double r444105 = t;
        double r444106 = r444103 + r444105;
        double r444107 = r444104 / r444106;
        double r444108 = r444104 - r444107;
        double r444109 = fma(r444108, r444108, r444104);
        double r444110 = r444103 / r444109;
        double r444111 = r444103 - r444110;
        return r444111;
}

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 2019156 +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)))))))))