Average Error: 0.0 → 0.0
Time: 4.3s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[\log \left(e^{1 - \frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\right)\]
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)}
↓
\log \left(e^{1 - \frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\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
 (log (exp (- 1.0 (/ 1.0 (+ (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t)) 6.0))))))
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 log(exp(1.0 - (1.0 / ((((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)) + 6.0))));
}

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

    \[\leadsto \color{blue}{1 - \frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\]
  3. Using strategy rm

  4. Applied add-log-exp_binary64_1170.5

    \[\leadsto 1 - \color{blue}{\log \left(e^{\frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\right)}\]

  5. Applied add-log-exp_binary64_1170.5

    \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\right)\]

  6. Applied diff-log_binary64_1700.8

    \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{e^{\frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}}\right)}\]

  7. Simplified0.0

    \[\leadsto \log \color{blue}{\left(e^{1 - \frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\right)}\]

  8. Final simplification0.0

    \[\leadsto \log \left(e^{1 - \frac{1}{\frac{\frac{4}{1 + t} + -8}{1 + t} + 6}}\right)\]

Reproduce

herbie shell --seed 2020354 
(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)))))))))