Average Error: 0.0 → 0.0
Time: 13.1s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}, 2 - \frac{2}{1 + t}, 2\right)}\]
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(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}, 2 - \frac{2}{1 + t}, 2\right)}
double f(double t) {
        double r570653 = 1.0;
        double r570654 = 2.0;
        double r570655 = t;
        double r570656 = r570654 / r570655;
        double r570657 = r570653 / r570655;
        double r570658 = r570653 + r570657;
        double r570659 = r570656 / r570658;
        double r570660 = r570654 - r570659;
        double r570661 = r570660 * r570660;
        double r570662 = r570654 + r570661;
        double r570663 = r570653 / r570662;
        double r570664 = r570653 - r570663;
        return r570664;
}
↓
double f(double t) {
        double r570665 = 1.0;
        double r570666 = 2.0;
        double r570667 = t;
        double r570668 = r570665 + r570667;
        double r570669 = r570666 / r570668;
        double r570670 = r570666 - r570669;
        double r570671 = cbrt(r570670);
        double r570672 = r570671 * r570671;
        double r570673 = r570672 * r570671;
        double r570674 = fma(r570673, r570670, r570666);
        double r570675 = r570665 / r570674;
        double r570676 = r570665 - r570675;
        return r570676;
}

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-cube-cbrt0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}}, 2 - \frac{2}{1 + t}, 2\right)}\]

  5. Final simplification0.0

    \[\leadsto 1 - \frac{1}{\mathsf{fma}\left(\left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot \sqrt[3]{2 - \frac{2}{1 + t}}\right) \cdot \sqrt[3]{2 - \frac{2}{1 + t}}, 2 - \frac{2}{1 + t}, 2\right)}\]

Reproduce

herbie shell --seed 2019153 +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)))))))))