Average Error: 0.0 → 0.0
Time: 2.4s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{{1}^{3} - {\left(\frac{1}{2 + {\left(2 - \frac{2}{1 + 1 \cdot t}\right)}^{2}}\right)}^{3}}{1 \cdot 1 + \frac{1}{2 + {\left(2 - \frac{2}{1 + 1 \cdot t}\right)}^{2}} \cdot \left(1 + \frac{1}{2 + {\left(2 - \frac{2}{1 + 1 \cdot t}\right)}^{2}}\right)}\]
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)}
↓
\frac{{1}^{3} - {\left(\frac{1}{2 + {\left(2 - \frac{2}{1 + 1 \cdot t}\right)}^{2}}\right)}^{3}}{1 \cdot 1 + \frac{1}{2 + {\left(2 - \frac{2}{1 + 1 \cdot t}\right)}^{2}} \cdot \left(1 + \frac{1}{2 + {\left(2 - \frac{2}{1 + 1 \cdot 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
 (/
  (-
   (pow 1.0 3.0)
   (pow (/ 1.0 (+ 2.0 (pow (- 2.0 (/ 2.0 (+ 1.0 (* 1.0 t)))) 2.0))) 3.0))
  (+
   (* 1.0 1.0)
   (*
    (/ 1.0 (+ 2.0 (pow (- 2.0 (/ 2.0 (+ 1.0 (* 1.0 t)))) 2.0)))
    (+ 1.0 (/ 1.0 (+ 2.0 (pow (- 2.0 (/ 2.0 (+ 1.0 (* 1.0 t)))) 2.0))))))))
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) (((double) pow(1.0, 3.0)) - ((double) pow((1.0 / ((double) (2.0 + ((double) pow(((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (1.0 * t))))))), 2.0))))), 3.0)))) / ((double) (((double) (1.0 * 1.0)) + ((double) ((1.0 / ((double) (2.0 + ((double) pow(((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (1.0 * t))))))), 2.0))))) * ((double) (1.0 + (1.0 / ((double) (2.0 + ((double) pow(((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (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. Simplified0.0

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

  4. Applied flip3--_binary640.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020210 
(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)))))))))