Average Error: 0.0 → 0.0
Time: 5.9s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{{2}^{3} + {\left(2 - \frac{2}{1 + t \cdot 1}\right)}^{6}} \cdot \left(\left(\left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)\right) \cdot \left(\left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)\right) \cdot 2\right)\right)\]
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 + t \cdot 1}\right)}^{6}} \cdot \left(\left(\left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)\right) \cdot \left(\left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)\right) + \left(2 \cdot 2 - \left(\left(2 - \frac{2}{1 + t \cdot 1}\right) \cdot \left(2 - \frac{2}{1 + t \cdot 1}\right)\right) \cdot 2\right)\right)
double code(double t) {
	return ((double) (1.0 - (1.0 / ((double) (2.0 + ((double) (((double) (2.0 - ((2.0 / t) / ((double) (1.0 + (1.0 / t)))))) * ((double) (2.0 - ((2.0 / t) / ((double) (1.0 + (1.0 / t)))))))))))));
}
↓
double code(double t) {
	return ((double) (1.0 - ((double) ((1.0 / ((double) (((double) pow(2.0, 3.0)) + ((double) pow(((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))), 6.0))))) * ((double) (((double) (((double) (((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))) * ((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))))) * ((double) (((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))) * ((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))))))) + ((double) (((double) (2.0 * 2.0)) - ((double) (((double) (((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))) * ((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (t * 1.0))))))))) * 2.0))))))))));
}

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

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

  4. Applied flip3-+0.0

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

  5. Applied associate-/r/0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020182 
(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)))))))))