Average Error: 0.5 → 0.4
Time: 13.2s
Precision: binary64
Cost: 20224
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}
(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (/ (/ (sqrt (* (* 2.0 PI) n)) (pow (* (* 2.0 PI) n) (/ k 2.0))) (sqrt k)))
double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}
↓
double code(double k, double n) {
	return (sqrt((2.0 * ((double) M_PI)) * n) / pow(((2.0 * ((double) M_PI)) * n), (k / 2.0))) / sqrt(k);
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.3
Cost98048
\[\frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}} \cdot \sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt[3]{\sqrt{k}} \cdot \sqrt[3]{\sqrt{k}}} \cdot \frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt[3]{\sqrt{k}}}\]
Alternative 2
Error1.1
Cost86144
\[\frac{\sqrt[3]{\sqrt{\left(2 \cdot \pi\right) \cdot n}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{\sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \cdot \frac{\frac{\sqrt[3]{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{\sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}}{\sqrt{k}}\]
Alternative 3
Error1.2
Cost85120
\[\frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}} \cdot \sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{\sqrt[3]{k}}}\]
Alternative 4
Error1.2
Cost85120
\[\frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}} \cdot \sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{\sqrt{k}}} \cdot \frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{\sqrt{k}}}\]
Alternative 5
Error1.1
Cost78848
\[\sqrt[3]{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{k}}} \cdot \left(\sqrt[3]{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{k}}} \cdot \sqrt[3]{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{k}}}\right)\]
Alternative 6
Error1.1
Cost65792
\[\left(\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}} \cdot \sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}\right) \cdot \frac{\sqrt[3]{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{k}}\]
Alternative 7
Error0.9
Cost65344
\[\frac{\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\left|\sqrt[3]{k}\right|} \cdot \frac{\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{\sqrt[3]{k}}}\]
Alternative 8
Error0.8
Cost65344
\[\frac{\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{\sqrt{k}}} \cdot \frac{\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{\sqrt{k}}}\]
Alternative 9
Error1.1
Cost52992
\[\left(\sqrt[3]{\sqrt{\left(2 \cdot \pi\right) \cdot n}} \cdot \sqrt[3]{\sqrt{\left(2 \cdot \pi\right) \cdot n}}\right) \cdot \frac{\frac{\sqrt[3]{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
Alternative 10
Error1.1
Cost52544
\[\frac{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt[3]{\sqrt{k}} \cdot \sqrt[3]{\sqrt{k}}} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt[3]{\sqrt{k}}}\]
Alternative 11
Error0.7
Cost52544
\[\sqrt{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{k}}}\]
Alternative 12
Error1.2
Cost52224
\[\frac{1}{\sqrt[3]{\sqrt{k}} \cdot \sqrt[3]{\sqrt{k}}} \cdot \frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt[3]{\sqrt{k}}}\]
Alternative 13
Error1.2
Cost52096
\[\frac{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt[3]{\sqrt{k}} \cdot \sqrt[3]{\sqrt{k}}}}{\sqrt[3]{\sqrt{k}}}\]
Alternative 14
Error0.7
Cost46016
\[\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}} \cdot \frac{\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}{\sqrt{k}}\]
Alternative 15
Error0.7
Cost40000
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{4}\right)}}{\sqrt{\sqrt{k}}} \cdot \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{4}\right)}}{\sqrt{\sqrt{k}}}\]
Alternative 16
Error0.6
Cost39808
\[\sqrt{\sqrt{\left(2 \cdot \pi\right) \cdot n}} \cdot \frac{\frac{\sqrt{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
Alternative 17
Error0.7
Cost39744
\[\frac{\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\frac{\sqrt{k}}{\sqrt{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}}\]
Alternative 18
Error0.8
Cost39616
\[\frac{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}{\left|\sqrt[3]{k}\right|} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\sqrt[3]{k}}}\]
Alternative 19
Error0.7
Cost39616
\[\frac{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\sqrt{k}}} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{\sqrt{k}}}\]
Alternative 20
Error0.9
Cost39296
\[\frac{1}{\left|\sqrt[3]{k}\right|} \cdot \frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{\sqrt[3]{k}}}\]
Alternative 21
Error0.7
Cost39296
\[\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{\sqrt{k}}} \cdot \frac{1}{\sqrt{\sqrt{k}}}\]
Alternative 22
Error0.9
Cost39168
\[\frac{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\left|\sqrt[3]{k}\right|}}{\sqrt{\sqrt[3]{k}}}\]
Alternative 23
Error0.7
Cost39168
\[\frac{\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{\sqrt{k}}}}{\sqrt{\sqrt{k}}}\]
Alternative 24
Error0.8
Cost33536
\[\frac{{\left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n} \cdot \left(\sqrt[3]{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt[3]{\left(2 \cdot \pi\right) \cdot n}\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 25
Error0.5
Cost33216
\[\frac{\sqrt{2 \cdot \pi}}{{\left(2 \cdot \pi\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{\frac{\sqrt{n}}{{n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
Alternative 26
Error19.5
Cost33088
\[\sqrt[3]{{\left(\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\right)}^{3}}\]
Alternative 27
Error0.8
Cost33024
\[\frac{{\left(\sqrt[3]{n} \cdot \left(\left(2 \cdot \pi\right) \cdot \left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right)\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 28
Error0.7
Cost32960
\[\frac{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 + \sqrt{k}\right)}\right)}^{\left(\frac{1 - \sqrt{k}}{2}\right)}}{\sqrt{k}}\]
Alternative 29
Error1.4
Cost32768
\[\frac{\frac{\sqrt{n}}{{n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \cdot \left(\sqrt{2} \cdot \sqrt{\pi}\right)\]
Alternative 30
Error3.4
Cost32640
\[e^{\log \left(\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}{\sqrt{k}}\right)}\]
Alternative 31
Error0.5
Cost26944
\[\frac{1}{{\left(2 \cdot \pi\right)}^{\left(\frac{k}{2}\right)}} \cdot \frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
Alternative 32
Error0.5
Cost26752
\[\frac{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 33
Error0.5
Cost26688
\[\frac{\frac{\sqrt{n}}{{n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}} \cdot {\left(2 \cdot \pi\right)}^{\left(0.5 - \frac{k}{2}\right)}\]
Alternative 34
Error0.6
Cost26624
\[\frac{\sqrt{2 \cdot \pi}}{\sqrt{k}} \cdot \frac{\sqrt{n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
Alternative 35
Error0.5
Cost26496
\[\frac{{\left(\sqrt{n} \cdot \left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 36
Error28.2
Cost26432
\[\frac{{\left(\sqrt[3]{8 \cdot {\left(\pi \cdot n\right)}^{3}}\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 37
Error28.5
Cost26368
\[\sqrt[3]{\frac{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}\right)}^{1.5}}{{k}^{1.5}}}\]
Alternative 38
Error3.3
Cost26368
\[\frac{{\left(e^{\log \left(\left(2 \cdot \pi\right) \cdot n\right)}\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 39
Error14.3
Cost26304
\[\frac{\sqrt[3]{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}\right)}^{1.5}}}{\sqrt{k}}\]
Alternative 40
Error0.6
Cost20800
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}{\frac{\sqrt{k}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{4}\right)}}}\]
Alternative 41
Error0.6
Cost20672
\[{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{4}\right)} \cdot \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{4}\right)}}{\sqrt{k}}\]
Alternative 42
Error0.6
Cost20288
\[\frac{{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)}}{\frac{\sqrt{k}}{{n}^{\left(\frac{1 - k}{2}\right)}}}\]
Alternative 43
Error0.4
Cost20288
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
Alternative 44
Error0.6
Cost20288
\[{\left(2 \cdot \pi\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{{n}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 45
Error0.4
Cost20224
\[\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\sqrt{k} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
Alternative 46
Error0.5
Cost19968
\[{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)} \cdot \sqrt{\frac{1}{k}}\]
Alternative 47
Error0.5
Cost19968
\[\frac{1}{\frac{\sqrt{k}}{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{\left(1 - k\right)}}}\]
Alternative 48
Error21.9
Cost19648
\[\frac{\sqrt{2} \cdot \sqrt{\pi \cdot n}}{\sqrt{k}}\]
Alternative 49
Error0.5
Cost13696
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
Alternative 50
Error0.5
Cost13568
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 51
Error60.5
Cost64
\[1\]
Alternative 52
Error41.8
Cost64
\[0\]
Alternative 53
Error62.8
Cost64
\[-1\]

Error

Derivation

  1. Initial program 0.5

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\]
  3. Using strategy rm

  4. Applied div-sub_binary64_7420.5

    \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}}\]

  5. Applied pow-sub_binary64_8130.4

    \[\leadsto \frac{\color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1}{2}\right)}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}}{\sqrt{k}}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]

  7. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}}\]

  8. Final simplification0.4

    \[\leadsto \frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))