Average Error: 0.0 → 0.0
Time: 13.9s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\left(\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) + 8} \cdot \left(\left(\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) - 2 \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right)\right) + 4\right)\]
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(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) + 8} \cdot \left(\left(\left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right) - 2 \cdot \left(\left(2 - \frac{2}{t + 1}\right) \cdot \left(2 - \frac{2}{t + 1}\right)\right)\right) + 4\right)
double f(double t) {
        double r839209 = 1.0;
        double r839210 = 2.0;
        double r839211 = t;
        double r839212 = r839210 / r839211;
        double r839213 = r839209 / r839211;
        double r839214 = r839209 + r839213;
        double r839215 = r839212 / r839214;
        double r839216 = r839210 - r839215;
        double r839217 = r839216 * r839216;
        double r839218 = r839210 + r839217;
        double r839219 = r839209 / r839218;
        double r839220 = r839209 - r839219;
        return r839220;
}
↓
double f(double t) {
        double r839221 = 1.0;
        double r839222 = 2.0;
        double r839223 = t;
        double r839224 = r839223 + r839221;
        double r839225 = r839222 / r839224;
        double r839226 = r839222 - r839225;
        double r839227 = r839226 * r839226;
        double r839228 = r839227 * r839226;
        double r839229 = r839228 * r839228;
        double r839230 = 8.0;
        double r839231 = r839229 + r839230;
        double r839232 = r839221 / r839231;
        double r839233 = r839227 * r839227;
        double r839234 = r839222 * r839227;
        double r839235 = r839233 - r839234;
        double r839236 = 4.0;
        double r839237 = r839235 + r839236;
        double r839238 = r839232 * r839237;
        double r839239 = r839221 - r839238;
        return r839239;
}

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

    \[\leadsto \color{blue}{1 - \frac{1}{2 + \left(2 - \frac{2}{1 + t}\right) \cdot \left(2 - \frac{2}{1 + t}\right)}}\]
  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. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 
(FPCore (t)
  :name "Kahan p13 Example 3"
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))