Average Error: 0.0 → 0.0
Time: 2.5s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\left(-{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot 2} \cdot \left(2 - \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)\]
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(-{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) + 2 \cdot 2} \cdot \left(2 - \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)
double f(double t) {
        double r44386 = 1.0;
        double r44387 = 2.0;
        double r44388 = t;
        double r44389 = r44387 / r44388;
        double r44390 = r44386 / r44388;
        double r44391 = r44386 + r44390;
        double r44392 = r44389 / r44391;
        double r44393 = r44387 - r44392;
        double r44394 = r44393 * r44393;
        double r44395 = r44387 + r44394;
        double r44396 = r44386 / r44395;
        double r44397 = r44386 - r44396;
        return r44397;
}
↓
double f(double t) {
        double r44398 = 1.0;
        double r44399 = 2.0;
        double r44400 = t;
        double r44401 = r44399 / r44400;
        double r44402 = r44398 / r44400;
        double r44403 = r44398 + r44402;
        double r44404 = r44401 / r44403;
        double r44405 = r44399 - r44404;
        double r44406 = 3.0;
        double r44407 = pow(r44405, r44406);
        double r44408 = -r44407;
        double r44409 = r44408 * r44405;
        double r44410 = r44399 * r44399;
        double r44411 = r44409 + r44410;
        double r44412 = r44398 / r44411;
        double r44413 = r44405 * r44405;
        double r44414 = r44399 - r44413;
        double r44415 = r44412 * r44414;
        double r44416 = r44398 - r44415;
        return r44416;
}

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 flip-+0.0

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

  4. Applied associate-/r/0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))