Difference between revisions of "User:Regina/MyDrafts"
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| − | + | ''' Accuracy ''' | |
| − | + | ||
| − | + | This article is about situations, where you think, that Calc calculates wrong. You might have found a bug, and then you should write an issue. But look here first; perhaps the unexpected results have reasons, you haven't been aware of. | |
| − | + | ||
| − | + | ||
| − | + | == Precision in Calc == | |
| − | + | Internally, Calc uses for its calculations floating point numbers in double precision defined in IEEE 754 standard. | |
| − | + | In default settings Calc rounds the displayed values to two decimals. You can get at most figures shown using the scientific format with format code <tt>0.00000000000000E+000</tt>. | |
| − | + | But although you can force Calc to show 15 decimal digits, these might not be all accurate. The following sections list some of the problems. | |
| − | : | + | ToDo: Explain setting "precision as shown". |
| − | === | + | == Converting between Number Systems == |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | Most non integer numbers have infinite decimal places in binary format, which have to be rounded somewhere. | |
| − | + | For example, the number 0.1 cannot be written exactly in binary format. Because of this rounding it can happen, that two mathematical different numbers have the same (rounded) internal binary value. | |
| − | == | + | The other way round, the decimal values are rounded to at most 15 figures. So you will see no difference between =SQRT(2) and =SQRT(2)+9E-15, but EXP(SQRT(2)) and EXP(SQRT(2)+9E-15) differs in the last shown figure. |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | == No Symbolic π == | |
| − | + | From mathematics you know <tt>sin(π)= 0</tt> and you know that <tt>tan(π/2)</tt> is undefined. But you cannot get these results in Calc, because the value π is always treated as rounded floating point number. It makes no difference using <tt>PI()</tt> or <tt>RADIANS(180)</tt>. Calc cannot evaluate π symbolically as computer algebra systems do. That is no special limitation of Calc, but other often used spreadsheet applications work only numerically, too. | |
| − | + | ||
| − | = | + | {| class="wikitable" |
| − | + | |- | |
| − | + | ! !! A !! B !! C | |
| − | + | |- | |
| − | + | ! 1 | |
| − | + | | 1.63317787283838E+016 || =TAN(PI()/2) || | |
| − | + | |- | |
| − | = | + | ! 2 |
| − | + | | 1.22460635382238E-016 || =SIN(RADIANS(180)) || | |
| − | + | |- | |
| − | + | ! 3 | |
| − | + | | || || | |
| − | + | |} | |
| − | === | + | == No Fractional Arithmetic == |
| − | + | ||
| − | + | ||
| + | ToDo: Explain why 1/7 + 1/3 results in 1/2, using format code # ?/? | ||
| + | == Cancellation == | ||
| − | + | If you subtract two non integer numbers, which have nearly the same value, the result has less significant figures then the initial values. | |
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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Cell A4 shows the correct result of <math>\scriptstyle 0.999411764795882-1699/1700</math>, calculated with a computer algebra system with high precision. | Cell A4 shows the correct result of <math>\scriptstyle 0.999411764795882-1699/1700</math>, calculated with a computer algebra system with high precision. | ||
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| − | |||
{| class="wikitable" | {| class="wikitable" | ||
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|- | |- | ||
! 1 | ! 1 | ||
| − | | 0.99999876543210000000 || | + | | 0.99999876543210000000 || || |
|- | |- | ||
! 2 | ! 2 | ||
| Line 135: | Line 82: | ||
|} | |} | ||
| − | + | Calculating manually gives 1−0.99999876543210000000 = 0.00000123456790000000 | |
| − | + | ||
| + | == Sensitivity == | ||
| + | |||
| + | Trigonometric functions are very sensitive for huge input values. That means, the results changes noticeable, if the input varies with one or two bit in the internal representation. | ||
| + | |||
| + | Example of formula input | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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|- | |- | ||
! 1 | ! 1 | ||
| − | | | + | | =2^44-0.004 || =sin(A1) || |
|- | |- | ||
! 2 | ! 2 | ||
| − | | | + | | =2^44 || =sin(A2) || |
|- | |- | ||
! 3 | ! 3 | ||
| + | | =2^44+0.004 || =sin(A3) || | ||
| + | |- | ||
| + | ! 4 | ||
| || || | | || || | ||
|} | |} | ||
| + | |||
| + | Results shown in 15 figures precision | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | ! !! A !! B !! C | ||
| + | |- | ||
| + | ! 1 | ||
| + | | 17592186044416.00 | ||
| + | || 0.386569623644289 | ||
| + | || | ||
| + | |- | ||
| + | ! 2 | ||
| + | | 17592186044416.00 | ||
| + | || 0.390169244205272 | ||
| + | || | ||
| + | |- | ||
| + | ! 3 | ||
| + | | 17592186044416.00 | ||
| + | || 0.393762911263629 | ||
| + | || | ||
| + | |- | ||
| + | ! 4 | ||
| + | | || || | ||
| + | |} | ||
| + | |||
| + | The correct value rounded to 15 figures precision would be sin(2^44) = 0.390169223351877 | ||
| + | |||
| + | The trigonometric functions are reliable for integral arguments up to 2^27 with at least 13 figures accuracy and for fractional arguments up to 2^27 with at least 8 figures. For larger values the accuracy decreases to about 6 figures accuracy for integral arguments and 2 figures accuracy for fractional values reaching 2^44. Huge arguments in the area above 2^44 might be totally wrong. If you need trigonometric functions for huge arguments, a spreadsheet is the wrong application for you. | ||
| + | |||
| + | The trigonometric functions are similar sensitive for arguments near their roots, but for those arguments the absolute result is near zero. So you will not notice the error, if you work with rounding to 15 or less decimal places. | ||
Latest revision as of 07:57, 25 May 2010
Accuracy
This article is about situations, where you think, that Calc calculates wrong. You might have found a bug, and then you should write an issue. But look here first; perhaps the unexpected results have reasons, you haven't been aware of.
Contents
Precision in Calc
Internally, Calc uses for its calculations floating point numbers in double precision defined in IEEE 754 standard.
In default settings Calc rounds the displayed values to two decimals. You can get at most figures shown using the scientific format with format code 0.00000000000000E+000.
But although you can force Calc to show 15 decimal digits, these might not be all accurate. The following sections list some of the problems.
ToDo: Explain setting "precision as shown".
Converting between Number Systems
Most non integer numbers have infinite decimal places in binary format, which have to be rounded somewhere. For example, the number 0.1 cannot be written exactly in binary format. Because of this rounding it can happen, that two mathematical different numbers have the same (rounded) internal binary value.
The other way round, the decimal values are rounded to at most 15 figures. So you will see no difference between =SQRT(2) and =SQRT(2)+9E-15, but EXP(SQRT(2)) and EXP(SQRT(2)+9E-15) differs in the last shown figure.
No Symbolic π
From mathematics you know sin(π)= 0 and you know that tan(π/2) is undefined. But you cannot get these results in Calc, because the value π is always treated as rounded floating point number. It makes no difference using PI() or RADIANS(180). Calc cannot evaluate π symbolically as computer algebra systems do. That is no special limitation of Calc, but other often used spreadsheet applications work only numerically, too.
| A | B | C | |
|---|---|---|---|
| 1 | 1.63317787283838E+016 | =TAN(PI()/2) | |
| 2 | 1.22460635382238E-016 | =SIN(RADIANS(180)) | |
| 3 |
No Fractional Arithmetic
ToDo: Explain why 1/7 + 1/3 results in 1/2, using format code # ?/?
Cancellation
If you subtract two non integer numbers, which have nearly the same value, the result has less significant figures then the initial values.
| A | B | C | |
|---|---|---|---|
| 1 | 9.99411764795882E-001 | =0.999411764795882 | |
| 2 | 9.99411764705882E-001 | =1699/1700 | |
| 3 | 8.99997854020285E-011 | =A1-A2 | |
| 4 | 8.99996470588235E-011 |
Cell A4 shows the correct result of 
, calculated with a computer algebra system with high precision.
| A | B | C | |
|---|---|---|---|
| 1 | 0.99999876543210000000 | ||
| 2 | 0.00000123456790002141 | =1-A1 | |
| 3 | |||
| 4 |
Calculating manually gives 1−0.99999876543210000000 = 0.00000123456790000000
Sensitivity
Trigonometric functions are very sensitive for huge input values. That means, the results changes noticeable, if the input varies with one or two bit in the internal representation.
Example of formula input
| A | B | C | |
|---|---|---|---|
| 1 | =2^44-0.004 | =sin(A1) | |
| 2 | =2^44 | =sin(A2) | |
| 3 | =2^44+0.004 | =sin(A3) | |
| 4 |
Results shown in 15 figures precision
| A | B | C | |
|---|---|---|---|
| 1 | 17592186044416.00 | 0.386569623644289 | |
| 2 | 17592186044416.00 | 0.390169244205272 | |
| 3 | 17592186044416.00 | 0.393762911263629 | |
| 4 |
The correct value rounded to 15 figures precision would be sin(2^44) = 0.390169223351877
The trigonometric functions are reliable for integral arguments up to 2^27 with at least 13 figures accuracy and for fractional arguments up to 2^27 with at least 8 figures. For larger values the accuracy decreases to about 6 figures accuracy for integral arguments and 2 figures accuracy for fractional values reaching 2^44. Huge arguments in the area above 2^44 might be totally wrong. If you need trigonometric functions for huge arguments, a spreadsheet is the wrong application for you.
The trigonometric functions are similar sensitive for arguments near their roots, but for those arguments the absolute result is near zero. So you will not notice the error, if you work with rounding to 15 or less decimal places.
