Average Error: 0.0 → 0.0
Time: 6.0s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)} \]
↓
\[e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}}\right)} \]
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)}
↓
e^{\mathsf{log1p}\left(\frac{-1}{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
 (exp
  (log1p (/ -1.0 (+ 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 exp(log1p((-1.0 / (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. Applied egg-rr0.0

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}}\right)}} \]

  3. Final simplification0.0

    \[\leadsto e^{\mathsf{log1p}\left(\frac{-1}{2 + {\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}^{2}}\right)} \]

Reproduce

herbie shell --seed 2022130 
(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)))))))))