Average Error: 0.0 → 0.0
Time: 6.1s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{4 - {\left(2 - \frac{2}{1 + t}\right)}^{4}} \cdot \left(2 - {\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}\right)\]
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}{4 - {\left(2 - \frac{2}{1 + t}\right)}^{4}} \cdot \left(2 - {\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}\right)
(FPCore (t)
 :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)))))))))
↓
(FPCore (t)
 :precision binary64
 (-
  1.0
  (*
   (/ 1.0 (- 4.0 (pow (- 2.0 (/ 2.0 (+ 1.0 t))) 4.0)))
   (- 2.0 (pow (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) 2.0)))))
double code(double t) {
	return 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)))))));
}
↓
double code(double t) {
	return 1.0 - ((1.0 / (4.0 - pow((2.0 - (2.0 / (1.0 + t))), 4.0))) * (2.0 - pow((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))), 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. Using strategy rm

  3. Applied pow2_binary64_1600.0

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

  5. Applied flip-+_binary64_530.0

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

  6. Applied associate-/r/_binary64_250.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020289 
(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)))))))))