Average Error: 0.0 → 0.0
Time: 12.8s
Precision: 64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{1 + \frac{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}{\mathsf{fma}\left(2, 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right)\right)} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \frac{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}{\mathsf{fma}\left(2, 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right)\right)} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\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)}
↓
\frac{1 + \frac{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}{\mathsf{fma}\left(2, 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right)\right)} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \frac{{2}^{3} - {\left(\frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{3}}{\mathsf{fma}\left(2, 2, \frac{\frac{2}{t}}{1 + \frac{1}{t}} \cdot \left(\frac{\frac{2}{t}}{1 + \frac{1}{t}} + 2\right)\right)} \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}
double code(double t) {
	return ((1.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))));
}
↓
double code(double t) {
	return ((1.0 + (((pow(2.0, 3.0) - pow(((2.0 / t) / (1.0 + (1.0 / t))), 3.0)) / fma(2.0, 2.0, (((2.0 / t) / (1.0 + (1.0 / t))) * (((2.0 / t) / (1.0 + (1.0 / t))) + 2.0)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + (((pow(2.0, 3.0) - pow(((2.0 / t) / (1.0 + (1.0 / t))), 3.0)) / fma(2.0, 2.0, (((2.0 / t) / (1.0 + (1.0 / t))) * (((2.0 / t) / (1.0 + (1.0 / t))) + 2.0)))) * (2.0 - ((2.0 / t) / (1.0 + (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

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

  3. Applied flip3--0.1

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

  4. Simplified0.1

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

  6. Applied flip3--0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))) (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t))))))))