Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{1 + \frac{1}{t}}\right)}\]
1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{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{\sqrt{2}}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{t}}}{1 + \frac{1}{t}}\right)}
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)))) * (2.0 - ((sqrt(2.0) / (cbrt(t) * cbrt(t))) * ((sqrt(2.0) / cbrt(t)) / (1.0 + (1.0 / t)))))))));
}

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 *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Applied times-frac0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (t)
  :name "Kahan p13 Example 3"
  :precision binary64
  (- 1 (/ 1 (+ 2 (* (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))) (- 2 (/ (/ 2 t) (+ 1 (/ 1 t)))))))))