Average Error: 0.0 → 0.0
Time: 9.9s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
↓
\frac{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 1\right)}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r1581752 = 1.0;
        double r1581753 = 2.0;
        double r1581754 = t;
        double r1581755 = r1581753 / r1581754;
        double r1581756 = r1581752 / r1581754;
        double r1581757 = r1581752 + r1581756;
        double r1581758 = r1581755 / r1581757;
        double r1581759 = r1581753 - r1581758;
        double r1581760 = r1581759 * r1581759;
        double r1581761 = r1581752 + r1581760;
        double r1581762 = r1581753 + r1581760;
        double r1581763 = r1581761 / r1581762;
        return r1581763;
}
↓
double f(double t) {
        double r1581764 = 2.0;
        double r1581765 = 1.0;
        double r1581766 = t;
        double r1581767 = r1581765 + r1581766;
        double r1581768 = r1581764 / r1581767;
        double r1581769 = r1581764 - r1581768;
        double r1581770 = fma(r1581769, r1581769, r1581765);
        double r1581771 = fma(r1581769, r1581769, r1581764);
        double r1581772 = r1581770 / r1581771;
        return r1581772;
}

Error

Bits error versus t

Derivation

  1. Initial program 0.0

    \[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))