Kahan p13 Example 3

Average Error: 0.0 → 0.0

Time: 1.7m

Precision: 64

Math

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

↓

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

C

double code(double t) {
	return ((double) (1.0 - ((double) (1.0 / ((double) (2.0 + ((double) (((double) (2.0 - ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))))) * ((double) (2.0 - ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t))))))))))))))));
}

↓

double code(double t) {
	return ((double) (1.0 - ((double) (1.0 / ((double) (2.0 + ((double) (((double) (2.0 - ((double) (((double) (2.0 / t)) / ((double) (1.0 + ((double) (1.0 / t)))))))) * ((double) (2.0 - ((double) (((double) (((double) (((double) (((double) cbrt(2.0)) * ((double) cbrt(2.0)))) / ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))) / ((double) (((double) cbrt(((double) (1.0 + ((double) (1.0 / t)))))) * ((double) cbrt(((double) (1.0 + ((double) (1.0 / t)))))))))) * ((double) (((double) (((double) cbrt(2.0)) / ((double) cbrt(t)))) / ((double) cbrt(((double) (1.0 + ((double) (1.0 / t))))))))))))))))))));
}

Error

Bits error versus t

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 0.0

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

4. Applied add-cube-cbrt 0.0

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

5. Applied add-cube-cbrt 0.0

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

6. Applied times-frac 0.0

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

7. Applied times-frac 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020113 
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))