Average Error: 0.0 → 0.0
Time: 2.6s
Precision: binary64
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \left(2 - \frac{2}{1 + 1 \cdot t}\right) \cdot \left(\sqrt[3]{2 - \frac{2}{1 + 1 \cdot t}} \cdot {\left(\sqrt[3]{2 - \frac{2}{1 + 1 \cdot t}}\right)}^{2}\right)}\]
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{2}{1 + 1 \cdot t}\right) \cdot \left(\sqrt[3]{2 - \frac{2}{1 + 1 \cdot t}} \cdot {\left(\sqrt[3]{2 - \frac{2}{1 + 1 \cdot 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 (+ 1.0 (* 1.0 t))))
     (*
      (cbrt (- 2.0 (/ 2.0 (+ 1.0 (* 1.0 t)))))
      (pow (cbrt (- 2.0 (/ 2.0 (+ 1.0 (* 1.0 t))))) 2.0)))))))
double code(double t) {
	return ((double) (1.0 - (1.0 / ((double) (2.0 + ((double) (((double) (2.0 - ((2.0 / t) / ((double) (1.0 + (1.0 / t)))))) * ((double) (2.0 - ((2.0 / t) / ((double) (1.0 + (1.0 / t)))))))))))));
}
↓
double code(double t) {
	return ((double) (1.0 - (1.0 / ((double) (2.0 + ((double) (((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (1.0 * t))))))) * ((double) (((double) cbrt(((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (1.0 * t))))))))) * ((double) pow(((double) cbrt(((double) (2.0 - (2.0 / ((double) (1.0 + ((double) (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 Error: 0.0 bits

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

  2. SimplifiedError: 0.0 bits

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

  4. Applied add-cube-cbrtError: 0.0 bits

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

  5. SimplifiedError: 0.0 bits

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

  6. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))))))))