Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 23.5s

Precision: 64

Math

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

↓

\[1 - \frac{1}{8 + {\left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right)}^{3}} \cdot \left(\left(\left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right) \cdot \left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right) - \left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right) \cdot 2\right) + 4\right)\]

C

double f(double t) {
        double r2302655 = 1.0;
        double r2302656 = 2.0;
        double r2302657 = t;
        double r2302658 = r2302656 / r2302657;
        double r2302659 = r2302655 / r2302657;
        double r2302660 = r2302655 + r2302659;
        double r2302661 = r2302658 / r2302660;
        double r2302662 = r2302656 - r2302661;
        double r2302663 = r2302662 * r2302662;
        double r2302664 = r2302656 + r2302663;
        double r2302665 = r2302655 / r2302664;
        double r2302666 = r2302655 - r2302665;
        return r2302666;
}

↓

double f(double t) {
        double r2302667 = 1.0;
        double r2302668 = 8.0;
        double r2302669 = 2.0;
        double r2302670 = t;
        double r2302671 = r2302667 + r2302670;
        double r2302672 = r2302669 / r2302671;
        double r2302673 = r2302669 - r2302672;
        double r2302674 = r2302673 * r2302673;
        double r2302675 = 3.0;
        double r2302676 = pow(r2302674, r2302675);
        double r2302677 = r2302668 + r2302676;
        double r2302678 = r2302667 / r2302677;
        double r2302679 = r2302674 * r2302674;
        double r2302680 = r2302674 * r2302669;
        double r2302681 = r2302679 - r2302680;
        double r2302682 = 4.0;
        double r2302683 = r2302681 + r2302682;
        double r2302684 = r2302678 * r2302683;
        double r2302685 = r2302667 - r2302684;
        return r2302685;
}

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. Simplified 0.0

   \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}}\]
3. Using strategy rm

4. Applied flip3-+ 0.0

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

5. Applied associate-/r/ 0.0

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

6. Final simplification 0.0

   \[\leadsto 1 - \frac{1}{8 + {\left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right)}^{3}} \cdot \left(\left(\left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right) \cdot \left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right) - \left(\left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)\right) \cdot 2\right) + 4\right)\]

Reproduce

herbie shell --seed 2019121 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))