Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\frac{\left(8 - \frac{8}{\left(\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}\right) \cdot \left(8 - \frac{8}{\left(\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}\right)}{\left(4 + \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 + \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + 2}\]
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}{\frac{\left(8 - \frac{8}{\left(\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}\right) \cdot \left(8 - \frac{8}{\left(\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)}\right)}{\left(4 + \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 + \left(2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} + 2}
double f(double t) {
        double r7634564 = 1.0;
        double r7634565 = 2.0;
        double r7634566 = t;
        double r7634567 = r7634565 / r7634566;
        double r7634568 = r7634564 / r7634566;
        double r7634569 = r7634564 + r7634568;
        double r7634570 = r7634567 / r7634569;
        double r7634571 = r7634565 - r7634570;
        double r7634572 = r7634571 * r7634571;
        double r7634573 = r7634565 + r7634572;
        double r7634574 = r7634564 / r7634573;
        double r7634575 = r7634564 - r7634574;
        return r7634575;
}
↓
double f(double t) {
        double r7634576 = 1.0;
        double r7634577 = 8.0;
        double r7634578 = t;
        double r7634579 = r7634576 / r7634578;
        double r7634580 = r7634576 + r7634579;
        double r7634581 = r7634578 * r7634580;
        double r7634582 = r7634581 * r7634581;
        double r7634583 = r7634582 * r7634581;
        double r7634584 = r7634577 / r7634583;
        double r7634585 = r7634577 - r7634584;
        double r7634586 = r7634585 * r7634585;
        double r7634587 = 4.0;
        double r7634588 = 2.0;
        double r7634589 = r7634588 / r7634578;
        double r7634590 = r7634589 / r7634580;
        double r7634591 = r7634588 + r7634590;
        double r7634592 = r7634591 * r7634590;
        double r7634593 = r7634587 + r7634592;
        double r7634594 = r7634593 * r7634593;
        double r7634595 = r7634586 / r7634594;
        double r7634596 = r7634595 + r7634588;
        double r7634597 = r7634576 / r7634596;
        double r7634598 = r7634576 - r7634597;
        return r7634598;
}

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. Using strategy rm

  3. Applied flip3--0.0

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

  4. Applied flip3--0.0

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

  5. Applied frac-times0.0

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

  6. Simplified0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\color{blue}{\left(8 - \frac{8}{\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)\right)}\right) \cdot \left(8 - \frac{8}{\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(\left(t \cdot \left(1 + \frac{1}{t}\right)\right) \cdot \left(t \cdot \left(1 + \frac{1}{t}\right)\right)\right)}\right)}}{\left(2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}\]

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))