Average Error: 0.0 → 0.0
Time: 13.3min
Precision: binary64
Cost: 3136
\[1 - \frac{1}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
↓
\[1 - \frac{1}{2 + \frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - \frac{\frac{4}{t \cdot \left(1 + \frac{1}{t}\right)}}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}\]
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 + \frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - \frac{\frac{4}{t \cdot \left(1 + \frac{1}{t}\right)}}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}
(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))))
      (- 4.0 (/ (/ 4.0 (* t (+ 1.0 (/ 1.0 t)))) (* 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 - ((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)))) * (4.0 - ((4.0 / (t * (1.0 + (1.0 / t)))) / (t * (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 / 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

Alternatives

Alternative 1
Error0.0
Cost1216
\[1 - \frac{1}{\left(\frac{4}{1 + t} + -8\right) \cdot \frac{1}{1 + t} + 6}\]
Alternative 2
Error0.0
Cost1088
\[1 - \frac{1}{6 + \frac{\frac{4}{1 + t} + -8}{1 + t}}\]
Alternative 3
Error1.3
Cost704
\[1 - \frac{1}{6 + \frac{-4}{1 + t}}\]
Alternative 4
Error0.5
Cost1090
\[\begin{array}{l} \mathbf{if}\;t \leq -0.7833328076668553:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.5709223458111528:\\ \;\;\;\;1 - \left(0.5 - t \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array}\]
Alternative 5
Error0.5
Cost1090
\[\begin{array}{l} \mathbf{if}\;t \leq -0.7833328076668553:\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{elif}\;t \leq 0.5709223458111528:\\ \;\;\;\;0.5 + t \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{0.2222222222222222}{t} + 0.16666666666666666\right)\\ \end{array}\]
Alternative 6
Error0.5
Cost648
\[\begin{array}{l} \mathbf{if}\;t \leq -0.7833328076668553 \lor \neg \left(t \leq 0.5709223458111528\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5 + t \cdot t\\ \end{array}\]
Alternative 7
Error0.6
Cost648
\[\begin{array}{l} \mathbf{if}\;t \leq -0.48768247176014906 \lor \neg \left(t \leq 0.6771604334576292\right):\\ \;\;\;\;0.8333333333333334 - \frac{0.2222222222222222}{t}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array}\]
Alternative 8
Error1.0
Cost706
\[\begin{array}{l} \mathbf{if}\;t \leq -0.33369853968687047:\\ \;\;\;\;0.8333333333333334\\ \mathbf{elif}\;t \leq 0.9958746963970585:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0.8333333333333334\\ \end{array}\]
Alternative 9
Error25.8
Cost64
\[0.5\]

Error

Time

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 flip--_binary64_530.0

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

  4. Applied associate-*l/_binary64_210.0

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

  5. Simplified0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\color{blue}{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - \frac{\frac{4}{\left(1 + \frac{1}{t}\right) \cdot t}}{\left(1 + \frac{1}{t}\right) \cdot t}\right)}}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}\]

  6. Simplified0.0

    \[\leadsto \color{blue}{1 - \frac{1}{2 + \frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - \frac{\frac{4}{\left(1 + \frac{1}{t}\right) \cdot t}}{\left(1 + \frac{1}{t}\right) \cdot t}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}}\]

  7. Final simplification0.0

    \[\leadsto 1 - \frac{1}{2 + \frac{\left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(4 - \frac{\frac{4}{t \cdot \left(1 + \frac{1}{t}\right)}}{t \cdot \left(1 + \frac{1}{t}\right)}\right)}{2 + \frac{\frac{2}{t}}{1 + \frac{1}{t}}}}\]

Reproduce

herbie shell --seed 2021040 
(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)))))))))