Average Error: 0.0 → 0.0
Time: 8.5s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\sqrt{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \sqrt{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}\right)}{\left(2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right) \cdot \left(2 \cdot 2 + \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2 \cdot \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)\right)}}\]
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 + \frac{\left({2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}\right) \cdot \left(\sqrt{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}} \cdot \sqrt{{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)}}
double code(double t) {
	return ((double) (1.0 - ((double) (1.0 / ((double) (2.0 + ((double) (((double) (2.0 - ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))))) * ((double) (2.0 - ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t))))))))))))))));
}
↓
double code(double t) {
	return ((double) (1.0 - ((double) (1.0 / ((double) (2.0 + ((double) (((double) (((double) (((double) pow(2.0, 3.0)) - ((double) pow(((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))), 3.0)))) * ((double) (((double) sqrt(((double) (((double) pow(2.0, 3.0)) - ((double) pow(((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))), 3.0)))))) * ((double) sqrt(((double) (((double) pow(2.0, 3.0)) - ((double) pow(((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))), 3.0)))))))))) / ((double) (((double) (((double) (2.0 * 2.0)) + ((double) (((double) (((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))) * ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))))) + ((double) (2.0 * ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))))))))) * ((double) (((double) (2.0 * 2.0)) + ((double) (((double) (((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))) * ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))))) + ((double) (2.0 * ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t))))))))))))))))))))));
}

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. Using strategy rm

  7. Applied add-sqr-sqrt0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020156 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1.0 (/ 1.0 (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))))))