Average Error: 0.0 → 0.0
Time: 2.0s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{\sqrt[3]{{\left(\frac{\frac{4}{1 + t} + -8}{1 + t}\right)}^{3}} + 6}\]
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}{\sqrt[3]{{\left(\frac{\frac{4}{1 + t} + -8}{1 + t}\right)}^{3}} + 6}
(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 (+ (cbrt (pow (/ (+ (/ 4.0 (+ 1.0 t)) -8.0) (+ 1.0 t)) 3.0)) 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 1.0 - (1.0 / (cbrt(pow((((4.0 / (1.0 + t)) + -8.0) / (1.0 + t)), 3.0)) + 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-cbrt-cube_binary640.0

    \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt[3]{\left(\frac{\frac{4}{1 + t} + -8}{1 + t} \cdot \frac{\frac{4}{1 + t} + -8}{1 + t}\right) \cdot \frac{\frac{4}{1 + t} + -8}{1 + t}}} + 6}\]

  5. Simplified0.0

    \[\leadsto 1 - \frac{1}{\sqrt[3]{\color{blue}{{\left(\frac{\frac{4}{1 + t} + -8}{1 + t}\right)}^{3}}} + 6}\]

  6. Final simplification0.0

    \[\leadsto 1 - \frac{1}{\sqrt[3]{{\left(\frac{\frac{4}{1 + t} + -8}{1 + t}\right)}^{3}} + 6}\]

Reproduce

herbie shell --seed 2020233 
(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)))))))))