Average Error: 0.0 → 0.0
Time: 10.3s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{{2}^{3} + {\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)}^{6}} \cdot \left(2 \cdot 2 + \left(\left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) \cdot \left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) - 2 \cdot \left(\frac{2 \cdot \left(2 - \frac{2}{\left(1 \cdot \left(t + 1\right)\right) \cdot \left(1 \cdot \left(t + 1\right)\right)}\right)}{2 + \frac{2}{1 \cdot \left(t + 1\right)}} \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right)\right)\right)\]
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}{{2}^{3} + {\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)}^{6}} \cdot \left(2 \cdot 2 + \left(\left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) \cdot \left(\left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right) \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right) - 2 \cdot \left(\frac{2 \cdot \left(2 - \frac{2}{\left(1 \cdot \left(t + 1\right)\right) \cdot \left(1 \cdot \left(t + 1\right)\right)}\right)}{2 + \frac{2}{1 \cdot \left(t + 1\right)}} \cdot \left(2 - \frac{2}{1 \cdot \left(t + 1\right)}\right)\right)\right)\right)
double f(double t) {
        double r25243 = 1.0;
        double r25244 = 2.0;
        double r25245 = t;
        double r25246 = r25244 / r25245;
        double r25247 = r25243 / r25245;
        double r25248 = r25243 + r25247;
        double r25249 = r25246 / r25248;
        double r25250 = r25244 - r25249;
        double r25251 = r25250 * r25250;
        double r25252 = r25244 + r25251;
        double r25253 = r25243 / r25252;
        double r25254 = r25243 - r25253;
        return r25254;
}
↓
double f(double t) {
        double r25255 = 1.0;
        double r25256 = 2.0;
        double r25257 = 3.0;
        double r25258 = pow(r25256, r25257);
        double r25259 = t;
        double r25260 = 1.0;
        double r25261 = r25259 + r25260;
        double r25262 = r25255 * r25261;
        double r25263 = r25256 / r25262;
        double r25264 = r25256 - r25263;
        double r25265 = 6.0;
        double r25266 = pow(r25264, r25265);
        double r25267 = r25258 + r25266;
        double r25268 = r25255 / r25267;
        double r25269 = r25256 * r25256;
        double r25270 = r25264 * r25264;
        double r25271 = r25270 * r25270;
        double r25272 = r25262 * r25262;
        double r25273 = r25256 / r25272;
        double r25274 = r25256 - r25273;
        double r25275 = r25256 * r25274;
        double r25276 = r25256 + r25263;
        double r25277 = r25275 / r25276;
        double r25278 = r25277 * r25264;
        double r25279 = r25256 * r25278;
        double r25280 = r25271 - r25279;
        double r25281 = r25269 + r25280;
        double r25282 = r25268 * r25281;
        double r25283 = r25255 - r25282;
        return r25283;
}

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

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

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

  8. Applied flip--0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020042 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))