Average Error: 0.0 → 0.0
Time: 4.0s
Precision: binary64
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\frac{-5 - \frac{\frac{4}{1 + t} + -8}{1 + t}}{-6 - \frac{\frac{4}{1 + t} + -8}{1 + t}}\]
\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{-5 - \frac{\frac{4}{1 + t} + -8}{1 + t}}{-6 - \frac{\frac{4}{1 + t} + -8}{1 + t}}
(FPCore (t)
 :precision binary64
 (/
  (+
   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))))))))
↓
(FPCore (t)
 :precision binary64
 (/
  (- -5.0 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t)))
  (- -6.0 (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t)))))
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 (-5.0 - (((4.0 / (1.0 + t)) + -8.0) / (1.0 + t))) / (-6.0 - (((4.0 / (1.0 + t)) + -8.0) / (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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 5}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6}}\]
  3. Using strategy rm

  4. Applied frac-2neg_binary64_11120.0

    \[\leadsto \color{blue}{\frac{-\left(\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 5\right)}{-\left(\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6\right)}}\]

  5. Simplified0.0

    \[\leadsto \frac{\color{blue}{-5 - \frac{\frac{4}{1 + t} + -8}{1 + t}}}{-\left(\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6\right)}\]

  6. Simplified0.0

    \[\leadsto \frac{-5 - \frac{\frac{4}{1 + t} + -8}{1 + t}}{\color{blue}{-6 - \frac{\frac{4}{1 + t} + -8}{1 + t}}}\]

  7. Final simplification0.0

    \[\leadsto \frac{-5 - \frac{\frac{4}{1 + t} + -8}{1 + t}}{-6 - \frac{\frac{4}{1 + t} + -8}{1 + t}}\]

Reproduce

herbie shell --seed 2020355 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 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))))))))