Average Error: 0.0 → 0.0
Time: 36.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(2 - \frac{2}{1 + t}, 2 - \frac{\sqrt{2}}{\frac{1 + t}{\sqrt{2}}}, 2\right)}\]
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{\sqrt{2}}{\frac{1 + t}{\sqrt{2}}}, 2\right)}
double f(double t) {
        double r962715 = 1.0;
        double r962716 = 2.0;
        double r962717 = t;
        double r962718 = r962716 / r962717;
        double r962719 = r962715 / r962717;
        double r962720 = r962715 + r962719;
        double r962721 = r962718 / r962720;
        double r962722 = r962716 - r962721;
        double r962723 = r962722 * r962722;
        double r962724 = r962716 + r962723;
        double r962725 = r962715 / r962724;
        double r962726 = r962715 - r962725;
        return r962726;
}
↓
double f(double t) {
        double r962727 = 1.0;
        double r962728 = 2.0;
        double r962729 = t;
        double r962730 = r962727 + r962729;
        double r962731 = r962728 / r962730;
        double r962732 = r962728 - r962731;
        double r962733 = sqrt(r962728);
        double r962734 = r962730 / r962733;
        double r962735 = r962733 / r962734;
        double r962736 = r962728 - r962735;
        double r962737 = fma(r962732, r962736, r962728);
        double r962738 = r962727 / r962737;
        double r962739 = r962727 - r962738;
        return r962739;
}

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. Using strategy rm

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-/l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +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)))))))))