Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot {\left(\sqrt[3]{2 - \frac{2}{1 + t}}\right)}^{2}\right)}\]
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}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(\sqrt[3]{2 - \frac{2}{1 + t}} \cdot {\left(\sqrt[3]{2 - \frac{2}{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
   (+
    2.0
    (*
     (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
     (*
      (cbrt (- 2.0 (/ 2.0 (+ 1.0 t))))
      (pow (cbrt (- 2.0 (/ 2.0 (+ 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 / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (cbrt(2.0 - (2.0 / (1.0 + t))) * pow(cbrt(2.0 - (2.0 / (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 add-cube-cbrt_binary640.0

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

  4. Simplified0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020232 
(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)))))))))