Average Error: 0.0 → 0.0

Time: 6.5s

Precision: binary64

Cost: 3264

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

↓

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

↓

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

(FPCore (t)
 :precision binary64
 (/
  (+
   1.0
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   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
   (*
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))
    (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))
  (+
   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 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (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 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))))) / (2.0 + ((2.0 - ((2.0 / t) / (1.0 + (1.0 / t)))) * (2.0 - ((2.0 / t) / (1.0 + (1.0 / t))))));
}

Error

The X axis uses an exponential scale — Bits error versus `t`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	1.1
Cost	22208

\[\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \sqrt[3]{\frac{2}{1 + t} + -4} \cdot \left(\frac{2}{1 + t} \cdot \left(\sqrt[3]{\frac{2}{1 + t} + -4} \cdot \sqrt[3]{\frac{2}{1 + t} + -4}\right)\right)}\]

Alternative 2
Error	0.0
Cost	14528

\[\log \left(e^{\frac{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 5}{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 6}}\right)\]

Alternative 3
Error	0.0
Cost	1984

\[\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6}\]

Alternative 4
Error	0.0
Cost	1856

\[\frac{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 5}{\frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right) + 6}\]

Alternative 5
Error	0.0
Cost	1856

\[\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 6}\]

Alternative 6
Error	0.0
Cost	1856

\[\frac{1}{\frac{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 6}{\frac{-8 + \frac{4}{1 + t}}{1 + t} + 5}}\]

Alternative 7
Error	0.0
Cost	1728

\[\frac{-5 - \frac{-8 + \frac{4}{1 + t}}{1 + t}}{-6 - \frac{-8 + \frac{4}{1 + t}}{1 + t}}\]

Alternative 8
Error	30.8
Cost	1600

\[\frac{-5 - \frac{-8 + \frac{4}{1 + t}}{1 + t}}{-6 - \left(\frac{-8}{t} + \frac{12}{t \cdot t}\right)}\]

Alternative 9
Error	31.1
Cost	1344

\[\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{6 + \frac{-8}{t}}\]

Alternative 10
Error	31.1
Cost	1216

\[\frac{-5 - \frac{-8 + \frac{4}{1 + t}}{1 + t}}{-6 - \frac{-8}{t}}\]

Alternative 11
Error	26.8
Cost	1088

\[\frac{5 + \frac{2}{1 + t} \cdot \left(\frac{2}{1 + t} + -4\right)}{2}\]

Alternative 12
Error	31.2
Cost	704

\[\left(0.8333333333333334 - \frac{0.2222222222222222}{t}\right) + \frac{0.037037037037037035}{t \cdot t}\]

Alternative 13
Error	26.3
Cost	64

\[0.8333333333333334\]

Alternative 14
Error	26.2
Cost	64

\[0.5\]

Alternative 15
Error	51.2
Cost	64

\[1\]

Alternative 16
Error	62.0
Cost	64

\[0\]

Alternative 17
Error	63.0
Cost	64

\[-1\]

Derivation

Initial program 0.0
\[\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]
Simplified0.0
\[\leadsto \color{blue}{\frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}}\]
Final simplification0.0
\[\leadsto \frac{1 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}{2 + \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right) \cdot \left(2 - \frac{\frac{2}{t}}{1 + \frac{1}{t}}\right)}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (t)
  :name "Kahan p13 Example 2"
  :precision binary64
  (/ (+ 1.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))))) (+ 2.0 (* (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t)))) (- 2.0 (/ (/ 2.0 t) (+ 1.0 (/ 1.0 t))))))))