Average Error: 19.3 → 1.1
Time: 10.7s
Precision: binary64
Cost: 52032
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (*
  (* c0 (fabs (/ (/ (cbrt A) (cbrt V)) (cbrt l))))
  (sqrt (/ (/ (cbrt A) (cbrt V)) (cbrt l)))))
double code(double c0, double A, double V, double l) {
	return c0 * sqrt(A / (V * l));
}
↓
double code(double c0, double A, double V, double l) {
	return (c0 * fabs((cbrt(A) / cbrt(V)) / cbrt(l))) * sqrt((cbrt(A) / cbrt(V)) / cbrt(l));
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.4
Cost116544
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}} \cdot \left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right)}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)}}\]
Alternative 2
Error33.5
Cost103680
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}} \cdot \left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right)}}{\sqrt[3]{{\ell}^{0.6666666666666666}} \cdot \sqrt[3]{\sqrt[3]{\ell}}}}\]
Alternative 3
Error33.2
Cost97088
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right) \cdot \sqrt[3]{e^{\log \left(\sqrt[3]{V}\right)}}}}{\sqrt[3]{\ell}}}\]
Alternative 4
Error1.3
Cost84288
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}} \cdot \left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right)}}{\sqrt[3]{\ell}}}\]
Alternative 5
Error1.3
Cost84288
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sqrt[3]{\ell}}\right)}}\]
Alternative 6
Error32.6
Cost71360
\[\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}} \cdot \left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}}\right|\right)\]
Alternative 7
Error32.6
Cost71360
\[\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}} \cdot \left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}}\right|\right)\]
Alternative 8
Error32.6
Cost71360
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\sqrt{\ell}} \cdot \sqrt[3]{\sqrt{\ell}}}}\]
Alternative 9
Error34.8
Cost65088
\[c0 \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt{V}}}{\sqrt[3]{\ell}}}\right)\]
Alternative 10
Error8.3
Cost52288
\[c0 \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{A}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\right)\]
Alternative 11
Error48.7
Cost52032
\[\sqrt{\frac{\frac{\sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{\frac{\sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}} \cdot c0\right)\]
Alternative 12
Error2.2
Cost52032
\[c0 \cdot \left(\left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\right)\]
Alternative 13
Error38.2
Cost45888
\[c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}}{\sqrt{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt{\ell}}}\right)\]
Alternative 14
Error35.5
Cost45632
\[c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt{V}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt{V}}}{\ell}}\right)\]
Alternative 15
Error20.0
Cost39872
\[\sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(\sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt[3]{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}}\right)\]
Alternative 16
Error20.0
Cost39616
\[c0 \cdot \left(\sqrt[3]{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(\sqrt[3]{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt[3]{\sqrt{\frac{A}{V \cdot \ell}}}\right)\right)\]
Alternative 17
Error12.4
Cost39232
\[c0 \cdot \left(\left|\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt[3]{\ell}}}\right)\]
Alternative 18
Error6.3
Cost39104
\[\left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\ell}}\]
Alternative 19
Error6.3
Cost39104
\[c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{V}}{\sqrt[3]{\ell}}}\right)\]
Alternative 20
Error5.0
Cost39104
\[c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\ell}}\right)\]
Alternative 21
Error14.1
Cost32832
\[c0 \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{A}{V}}{\sqrt[3]{\ell}}}\right)\]
Alternative 22
Error38.3
Cost32704
\[\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\]
Alternative 23
Error37.4
Cost32704
\[c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\right)\]
Alternative 24
Error19.7
Cost26816
\[c0 \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \left(\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}\right)}\]
Alternative 25
Error38.3
Cost26560
\[\sqrt{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}}\]
Alternative 26
Error19.7
Cost26432
\[\left(c0 \cdot \left|\sqrt[3]{\frac{A}{V \cdot \ell}}\right|\right) \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}\]
Alternative 27
Error19.5
Cost26432
\[\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)\]
Alternative 28
Error19.7
Cost26432
\[c0 \cdot \left(\left|\sqrt[3]{\frac{A}{V \cdot \ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}}\right)\]
Alternative 29
Error19.5
Cost26432
\[c0 \cdot \left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)\]
Alternative 30
Error15.5
Cost26304
\[\left(c0 \cdot \left|\sqrt[3]{\frac{A}{V}}\right|\right) \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}}\]
Alternative 31
Error20.1
Cost26304
\[\left(\sqrt[3]{c0} \cdot \sqrt[3]{c0}\right) \cdot \left(\sqrt[3]{c0} \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\right)\]
Alternative 32
Error14.4
Cost26304
\[c0 \cdot \left(\left|\sqrt[3]{\frac{A}{V}}\right| \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}}\right)\]
Alternative 33
Error18.6
Cost26304
\[c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V} \cdot \frac{\sqrt[3]{A}}{\ell}}\]
Alternative 34
Error13.2
Cost26176
\[c0 \cdot \left(\left|\sqrt[3]{A}\right| \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{V}}{\ell}}\right)\]
Alternative 35
Error51.3
Cost26176
\[\left(c0 \cdot \sqrt{\frac{\sqrt{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt{A}}{\ell}}\]
Alternative 36
Error50.8
Cost26176
\[c0 \cdot \left(\sqrt{\frac{\sqrt{A}}{V}} \cdot \sqrt{\frac{\sqrt{A}}{\ell}}\right)\]
Alternative 37
Error39.8
Cost26176
\[\left(c0 \cdot \sqrt{\sqrt{A}}\right) \cdot \sqrt{\frac{\frac{\sqrt{A}}{V}}{\ell}}\]
Alternative 38
Error48.6
Cost26048
\[c0 \cdot \left|\frac{\frac{\sqrt{A}}{\sqrt{V}}}{\sqrt{\ell}}\right|\]
Alternative 39
Error42.2
Cost19776
\[\sqrt{c0} \cdot \left(\sqrt{\frac{A}{V \cdot \ell}} \cdot \sqrt{c0}\right)\]
Alternative 40
Error41.4
Cost19776
\[c0 \cdot \sqrt{\frac{\sqrt{A}}{V} \cdot \frac{\sqrt{A}}{\ell}}\]
Alternative 41
Error41.6
Cost19712
\[\sqrt[3]{{\left(c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\right)}^{3}}\]
Alternative 42
Error39.9
Cost19712
\[c0 \cdot \sqrt{\sqrt[3]{{\left(\frac{A}{V \cdot \ell}\right)}^{3}}}\]
Alternative 43
Error28.1
Cost19712
\[c0 \cdot \sqrt[3]{{\left(\sqrt{\frac{A}{V \cdot \ell}}\right)}^{3}}\]
Alternative 44
Error28.1
Cost19712
\[c0 \cdot \sqrt[3]{{\left(\sqrt{\frac{\frac{A}{V}}{\ell}}\right)}^{3}}\]
Alternative 45
Error21.7
Cost19648
\[c0 \cdot e^{\log \left(\sqrt{\frac{A}{V \cdot \ell}}\right)}\]
Alternative 46
Error37.8
Cost13376
\[c0 \cdot \left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)\]
Alternative 47
Error38.0
Cost13376
\[c0 \cdot \left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right)\]
Alternative 48
Error38.1
Cost13376
\[c0 \cdot \left(\sqrt{\frac{1}{V}} \cdot \sqrt{\frac{A}{\ell}}\right)\]
Alternative 49
Error39.0
Cost13376
\[\left(\sqrt{A} \cdot c0\right) \cdot \sqrt{\frac{1}{V \cdot \ell}}\]
Alternative 50
Error39.0
Cost13376
\[\sqrt{\frac{A}{\ell}} \cdot \left(c0 \cdot \sqrt{\frac{1}{V}}\right)\]
Alternative 51
Error38.8
Cost13376
\[\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}\]
Alternative 52
Error38.8
Cost13248
\[\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\]
Alternative 53
Error37.8
Cost13248
\[c0 \cdot \frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}}\]
Alternative 54
Error38.8
Cost13248
\[\frac{\sqrt{A} \cdot c0}{\sqrt{V \cdot \ell}}\]
Alternative 55
Error37.8
Cost13248
\[c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\]
Alternative 56
Error19.8
Cost6976
\[c0 \cdot \sqrt{\frac{1}{\frac{\ell}{\frac{A}{V}}}}\]
Alternative 57
Error19.4
Cost6976
\[c0 \cdot \sqrt{A \cdot \frac{\frac{1}{V}}{\ell}}\]
Alternative 58
Error19.6
Cost6976
\[c0 \cdot \sqrt{A \cdot \frac{1}{V \cdot \ell}}\]
Alternative 59
Error19.6
Cost6976
\[c0 \cdot \sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}\]
Alternative 60
Error19.3
Cost6976
\[c0 \cdot \sqrt{\frac{1}{V} \cdot \frac{A}{\ell}}\]
Alternative 61
Error19.4
Cost6848
\[c0 \cdot \sqrt{\frac{\frac{A}{V}}{\ell}}\]
Alternative 62
Error19.3
Cost6848
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
Alternative 63
Error64.0
Cost576
\[c0 \cdot \left(\frac{A}{V \cdot \ell} \cdot \mathsf{NaN}\right)\]
Alternative 64
Error64.0
Cost448
\[c0 \cdot \frac{\mathsf{NaN}}{V \cdot \ell}\]
Alternative 65
Error61.7
Cost64
\[1\]
Alternative 66
Error54.6
Cost64
\[0\]
Alternative 67
Error61.7
Cost64
\[-1\]

Error

Derivation

  1. Initial program 19.3

    \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
  2. Using strategy rm

  3. Applied associate-/r*_binary64_138619.4

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt_binary64_147719.8

    \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

  6. Applied add-cube-cbrt_binary64_147719.9

    \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

  7. Applied add-cube-cbrt_binary64_147720.0

    \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}}}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

  8. Applied times-frac_binary64_144820.0

    \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

  9. Applied times-frac_binary64_144815.7

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}\]

  10. Applied sqrt-prod_binary64_14587.4

    \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\right)}\]

  11. Applied associate-*r*_binary64_13827.4

    \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}\]

  12. Simplified1.1

    \[\leadsto \color{blue}{\left(\left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right| \cdot c0\right)} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

  13. Simplified1.1

    \[\leadsto \color{blue}{\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}\]

  14. Final simplification1.1

    \[\leadsto \left(c0 \cdot \left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

Reproduce

herbie shell --seed 2021022 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))