Average Error: 0.0 → 0.0
Time: 19.7s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)\]
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}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)
double f(double t) {
        double r30612 = 1.0;
        double r30613 = 2.0;
        double r30614 = t;
        double r30615 = r30613 / r30614;
        double r30616 = r30612 / r30614;
        double r30617 = r30612 + r30616;
        double r30618 = r30615 / r30617;
        double r30619 = r30613 - r30618;
        double r30620 = r30619 * r30619;
        double r30621 = r30613 + r30620;
        double r30622 = r30612 / r30621;
        double r30623 = r30612 - r30622;
        return r30623;
}
↓
double f(double t) {
        double r30624 = 1.0;
        double r30625 = 2.0;
        double r30626 = t;
        double r30627 = r30626 * r30624;
        double r30628 = r30627 + r30624;
        double r30629 = r30625 / r30628;
        double r30630 = r30625 - r30629;
        double r30631 = 6.0;
        double r30632 = pow(r30630, r30631);
        double r30633 = 3.0;
        double r30634 = pow(r30625, r30633);
        double r30635 = r30632 + r30634;
        double r30636 = r30624 / r30635;
        double r30637 = r30630 * r30630;
        double r30638 = r30637 * r30637;
        double r30639 = r30625 * r30625;
        double r30640 = r30637 * r30625;
        double r30641 = r30639 - r30640;
        double r30642 = r30638 + r30641;
        double r30643 = r30636 * r30642;
        double r30644 = r30624 - r30643;
        return r30644;
}

Error

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

    \[\leadsto 1 - \color{blue}{\frac{1}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)\]

  7. Final simplification0.0

    \[\leadsto 1 - \frac{1}{{\left(2 - \frac{2}{t \cdot 1 + 1}\right)}^{6} + {2}^{3}} \cdot \left(\left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{t \cdot 1 + 1}\right) \cdot \left(2 - \frac{2}{t \cdot 1 + 1}\right)\right) \cdot 2\right)\right)\]

Reproduce

herbie shell --seed 2019304 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))