Editing Rounding (section)

==Rounding to integer==
The most basic form of rounding is to replace an arbitrary number by an integer. All the following rounding modes are concrete implementations of an abstract single-argument "round()" procedure. These are true functions (with the exception of those that use randomness).

===Directed rounding to an integer===
These four methods are called '''directed rounding to an integer''', as the displacements from the original number {{mvar|x}} to the rounded value {{mvar|y}} are all directed toward or away from the same limiting value (0, [[extended real number line|+∞]], or −∞). Directed rounding is used in [[interval arithmetic]] and is often required in financial calculations.

If {{mvar|x}} is positive, round-down is the same as round-toward-zero, and round-up is the same as round-away-from-zero. If {{mvar|x}} is negative, round-down is the same as round-away-from-zero, and round-up is the same as round-toward-zero. In any case, if {{mvar|x}} is an integer, {{mvar|y}} is just {{mvar|x}}.

Where many calculations are done in sequence, the choice of rounding method can have a very significant effect on the result. A famous instance involved a new [[stock index|index]] set up by the [[Vancouver Stock Exchange]] in 1982. It was initially set at 1000.000 (three decimal places of accuracy), and after 22 months had fallen to about 520, although the market appeared to be rising. The problem was caused by the index being recalculated thousands of times daily, and always being truncated (rounded down) to 3 decimal places, in such a way that the rounding errors accumulated. Recalculating the index for the same period using rounding to the nearest thousandth rather than truncation corrected the index value from 524.811 up to 1098.892.<ref>
{{cite book |title=Accuracy and stability of numerical algorithms  |author-first=Nicholas John |author-last=Higham |author-link=Nicholas Higham |page=54 |isbn=978-0-89871-521-7 |date=2002| url=https://books.google.com/books?id=7J52J4GrsJkC&dq=vancouver+stock+exchange+rounding&pg=PA54 |doi=10.1137/1.9780898718027.ch2 |edition=2nd |url-access=limited}} {{pb}} {{cite journal |last=Nievergelt |first=Yves |year=2000 |title=Rounding Errors to Knock Your Stocks Off |journal=Mathematics Magazine |volume=73 |issue=1 |pages=47–48 |doi=10.1080/0025570X.2000.11996800 |jstor=2691491 }} {{pb}} {{cite news |last=Quinn |first=Kevin |title=Ever had problems rounding off figures? This stock exchange has |newspaper=Wall Street Journal |date=8 November 1983 |url=https://www5.in.tum.de/~huckle/Vancouv.pdf}} {{pb}} {{cite news |last=Lilley |first=Wayne |newspaper=The Toronto Star |date=November 29, 1983 |title=Vancouver stock index has right number at last |url=https://www5.in.tum.de/~huckle/Vancouv.pdf }} </ref>

For the examples below, {{math|sgn(''x'')}} refers to the [[sign function]] applied to the original number, {{mvar|x}}.

====Rounding down====
One may '''round down''' (or take the '''[[floor and ceiling functions|floor]]''', or '''round toward negative infinity'''): {{mvar|y}} is the largest integer that does not exceed {{mvar|x}}.

: <math>y = \mathrm{floor}(x) = \left\lfloor x \right\rfloor = -\left\lceil -x \right\rceil</math>

For example, 23.7 gets rounded to 23, and −23.2 gets rounded to −24.

====Rounding up====
One may also '''round up''' (or take the '''[[floor and ceiling functions|ceiling]]''', or '''round toward positive infinity'''): {{mvar|y}} is the smallest integer that is not less than {{mvar|x}}.

: <math>y = \operatorname{ceil}(x) = \left\lceil x \right\rceil = -\left\lfloor -x \right\rfloor</math>

For example, 23.2 gets rounded to 24, and −23.7 gets rounded to −23.

====Rounding toward zero====
One may also '''round toward zero''' (or '''[[truncation|truncate]]''', or '''round away from infinity'''): {{mvar|y}} is the integer that is closest to {{mvar|x}} such that it is between 0 and {{mvar|x}} (included); i.e. {{mvar|y}} is the integer part of {{mvar|x}}, without its fraction digits.

: <math>
y = \operatorname{truncate}(x)
= \sgn(x) \left\lfloor \left| x \right| \right\rfloor
= -\sgn(x) \left\lceil -\left| x \right| \right\rceil
= \begin{cases}
  \left\lfloor x \right\rfloor & x \ge 0 \\[5mu]
  \left\lceil x \right\rceil   & x < 0
\end{cases}
</math>

For example, 23.7 gets rounded to 23, and −23.7 gets rounded to −23.

====Rounding away from zero====
One may also '''round away from zero''' (or '''round toward infinity'''): {{mvar|y}} is the integer that is closest to 0 (or equivalently, to {{mvar|x}}) such that {{mvar|x}} is between 0 and {{mvar|y}} (included).

: <math>
y = \sgn(x) \left\lceil \left| x \right| \right\rceil
= -\sgn(x) \left\lfloor -\left| x \right| \right\rfloor
= \begin{cases}
  \left\lceil x \right\rceil   & x \ge 0 \\[5mu]
  \left\lfloor x \right\rfloor & x < 0
\end{cases}
</math>

For example, 23.2 gets rounded to 24, and −23.2 gets rounded to −24.

===Rounding to the nearest integer===
These six methods are called '''rounding to the nearest integer'''. Rounding a number {{mvar|x}} to the nearest integer requires some tie-breaking rule for those cases when {{mvar|x}} is exactly half-way between two integers – that is, when the fraction part of {{mvar|x}} is exactly 0.5.

If it were not for the 0.5 fractional parts, the round-off errors introduced by the round to nearest method would be symmetric: for every fraction that gets rounded down (such as 0.268), there is a complementary fraction (namely, 0.732) that gets rounded up by the same amount.

When rounding a large set of [[fixed-point arithmetic|fixed-point]] numbers with [[discrete uniform distribution|uniformly distributed]] fractional parts, the rounding errors by all values, with the omission of those having 0.5 fractional part, would statistically compensate each other. This means that the [[expected value|expected (average) value]] of the rounded numbers is equal to the expected value of the original numbers when numbers with fractional part 0.5 from the set are removed.

In practice, [[floating-point arithmetic|floating-point]] numbers are typically used, which have even more computational nuances because they are not equally spaced.

====Rounding half up====
One may '''round half up''' (or '''round half toward positive infinity'''), a tie-breaking rule that is widely used in many disciplines.{{citation needed|reason=This is surprising. A confusion with ''round half away from zero''?|date=June 2017}} That is, half-way values of {{mvar|x}} are always rounded up. If the fractional part of {{mvar|x}} is exactly 0.5, then {{math|1=''y'' = ''x'' + 0.5}}

: <math>y = \left\lfloor x + \tfrac12 \right\rfloor = -\left\lceil -x - \tfrac12 \right\rceil = \left\lceil \tfrac12 \lfloor 2x \rfloor \right\rceil</math>

For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −23.

Some programming languages (such as Java and Python) use "half up" to refer to ''[[#Rounding half away from zero|round half away from zero]]'' rather than ''round half toward positive infinity''.<ref name="java">{{cite web |title=java.math.RoundingMode |url=https://docs.oracle.com/javase/8/docs/api/java/math/RoundingMode.html |publisher=Oracle}}</ref><ref name="python">{{cite web |title=decimal – Decimal fixed point and floating point arithmetic |url=https://docs.python.org/3/library/decimal.html#rounding-modes |publisher=Python Software Foundation}}</ref>

This method only requires checking one digit to determine rounding direction in [[two's complement]] and similar representations.

====Rounding half down====
One may also '''round half down''' (or '''round half toward negative infinity''') as opposed to the more common ''round half up''. If the fractional part of {{mvar|x}} is exactly 0.5, then {{math|1=''y'' = ''x'' − 0.5}}

: <math>y = \left\lceil x - \tfrac12 \right\rceil = -\left\lfloor -x + \tfrac12 \right\rfloor = \left\lfloor \tfrac12 \lceil 2x \rceil \right\rfloor</math>

For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −24.

Some programming languages (such as Java and Python) use "half down" to refer to ''[[#Rounding half toward zero|round half toward zero]]'' rather than ''round half toward negative infinity''.<ref name="java"/><ref name="python"/>

====Rounding half toward zero====
One may also '''round half toward zero''' (or '''round half away from infinity''') as opposed to the conventional ''round half away from zero''. If the fractional part of {{mvar|x}} is exactly 0.5, then {{math|1=''y'' = ''x'' − 0.5}} if {{mvar|x}} is positive, and {{math|1=''y'' = ''x'' + 0.5}} if {{mvar|x}} is negative.

: <math>
y = \sgn(x) \left\lceil \left| x \right| - \tfrac12 \right\rceil
= -\sgn(x) \left\lfloor -\left| x \right| + \tfrac12 \right\rfloor
= \begin{cases}
  \left\lceil x - \tfrac12 \right\rceil = \left\lfloor \tfrac12 \lceil 2x \rceil \right\rfloor & x \ge 0 \\[5mu]
  \left\lfloor x + \tfrac12 \right\rfloor = \left\lceil \tfrac12 \lfloor 2x \rfloor \right\rceil & x < 0
\end{cases}
</math>

For example, 23.5 gets rounded to 23, and −23.5 gets rounded to −23.

This method treats positive and negative values symmetrically, and therefore is free of overall positive/negative bias if the original numbers are positive or negative with equal probability. It does, however, still have bias toward zero.

====Rounding half away from zero====
One may also '''round half away from zero''' (or '''round half toward infinity'''), a tie-breaking rule that is commonly taught and used, namely: If the fractional part of {{mvar|x}} is exactly 0.5, then {{math|1=''y'' = ''x'' + 0.5}} if {{mvar|x}} is positive, and {{math|1=''y'' = ''x'' − 0.5}} if {{mvar|x}} is negative.

: <math>
y = \sgn(x) \left\lfloor \left| x \right| + \tfrac12 \right\rfloor
= -\sgn(x) \left\lceil -\left| x \right| - \tfrac12 \right\rceil
= \begin{cases}
  \left\lfloor x + \tfrac12 \right\rfloor = \left\lceil \tfrac12 \lfloor 2x \rfloor \right\rceil & x \ge 0 \\[5mu]
  \left\lceil x - \tfrac12 \right\rceil = \left\lfloor \tfrac12 \lceil 2x \rceil \right\rfloor & x < 0
\end{cases}
</math>

For example, 23.5 gets rounded to 24, and −23.5 gets rounded to −24.

This can be more efficient on computers that use [[sign-magnitude]] representation for the values to be rounded, because only the first omitted digit needs to be considered to determine if it rounds up or down. This is one method used when rounding to [[significant figures]] due to its simplicity.

This method, also known as '''commercial rounding''',{{citation needed|date=November 2020}} treats positive and negative values symmetrically, and therefore is free of overall positive/negative bias if the original numbers are positive or negative with equal probability. It does, however, still have bias away from zero.

It is often used for currency conversions and price roundings (when the amount is first converted into the smallest significant subdivision of the currency, such as cents of a euro) as it is easy to explain by just considering the first fractional digit, independently of supplementary precision digits or sign of the amount (for strict equivalence between the paying and recipient of the amount).

====Rounding half to even====
One may also '''round half to even''', a tie-breaking rule without positive/negative bias ''and'' without bias toward/away from zero. By this convention, if the fractional part of {{mvar|x}} is 0.5, then {{mvar|y}} is the even integer nearest to {{mvar|x}}. Thus, for example, 23.5 becomes 24, as does 24.5; however, −23.5 becomes −24, as does −24.5. This function minimizes the expected error when summing over rounded figures, even when the inputs are mostly positive or mostly negative, provided they are neither mostly even nor mostly odd.

This variant of the round-to-nearest method is also called '''convergent rounding''', '''statistician's rounding''', '''Dutch rounding''', '''Gaussian rounding''', '''odd–even rounding''',<ref>[https://web.archive.org/web/20150603082624/http://mscweb.gsfc.nasa.gov/543web/files/GSFC-X-673-64-1F.pdf Engineering Drafting Standards Manual] (NASA), X-673-64-1F, p90</ref> or '''bankers' rounding'''.<ref>{{cite book |title=Postcards 4 Language Booster: Workbook with Grammar Builder |last1=Abbs |first1=Brian |last2=Barker |first2=Chris |last3=Freebairn |first3=Ingrid |publisher=Pearson Education |year=2003 |isbn=0-13-093904-8 |at=[https://books.google.com/books?id=mpvSnBZTsOgC&dq=%22bankers%20rounding%22&pg=PA85 {{pgs|85}}] |quote=Rounding to the nearest even number is also called 'bankers rounding' because the banks use this technique as well.}} {{pb}} {{cite book |publisher=Microsoft Corporation |title=Microsoft Pascal Compiler for the MS-DOS Operating System User's Guide |year=1985 |at=[https://archive.org/details/microsoft-pascal-3.30-users-guide/page/165/ {{pgs|165}}] |quote=Bankers' rounding is used when truncating real numbers that end with .5; that is, odd numbers are rounded up to an even integer, even numbers are rounded down to an even integer.}}</ref>

This is the default rounding mode used in [[IEEE 754]] operations for results in binary floating-point formats.

By eliminating bias, repeated addition or subtraction of independent numbers, as in a [[Random walk#One-dimensional random walk|one-dimensional random walk]], will give a rounded result with an error that tends to grow in proportion to the square root of the number of operations rather than linearly.

However, this rule distorts the distribution by increasing the probability of evens relative to odds. Typically this is less important{{citation needed|date=December 2021}} than the biases that are eliminated by this method.

====Rounding half to odd====
One may also '''round half to odd''', a similar tie-breaking rule to round half to even. In this approach, if the fractional part of {{mvar|x}} is 0.5, then {{mvar|y}} is the odd integer nearest to {{mvar|x}}. Thus, for example, 23.5 becomes 23, as does 22.5; while −23.5 becomes −23, as does −22.5.

This method is also free from positive/negative bias and bias toward/away from zero, provided the numbers to be rounded are neither mostly even nor mostly odd.  It also shares the round half to even property of distorting the original distribution, as it increases the probability of odds relative to evens. It was the method used for bank balances in the [[United Kingdom]] when it decimalized its currency<ref>Schedule 1 of the Decimal Currency Act 1969</ref>{{clarify|reason=Needs a complete reference.|date=May 2023}}.

This variant is almost never used in computations, except in situations where one wants to avoid increasing the scale of floating-point numbers, which have a limited exponent range. With ''round half to even'', a non-infinite number would round to infinity, and a small {{em|[[denormal]]}} value would round to a normal non-zero value. Effectively, this mode prefers preserving the existing scale of tie numbers, avoiding out-of-range results when possible for numeral systems of even [[radix]] (such as binary and decimal).{{clarify|reason=The ''problem'' would be avoided only for halfway numbers. So, how can this be useful in practice? Any practical example?|post-text=(see [[Talk:Rounding#Round half to odd|talk]])|date=September 2017}}.

===Rounding to prepare for shorter precision===
This rounding mode is used to avoid getting a potentially wrong result after [[#Double rounding|multiple roundings]]. This can be achieved if all roundings except the final one are done using rounding to prepare for shorter precision ("RPSP"), and only the final rounding uses the externally requested mode.

With decimal arithmetic, final digits of 0 and 5 are avoided; if there is a choice between numbers with the least significant digit 0 or 1, 4 or 5, 5 or 6, 9 or 0, then the digit different from 0 or 5 shall be selected; otherwise, the choice is arbitrary. IBM defines that, in the latter case, a digit with the smaller magnitude shall be selected.<ref>[https://www.ibm.com/support/pages/zarchitecture-principles-operation IBM z/Architecture Principles of Operation]</ref> RPSP can be applied with the step between two consequent roundings as small as a single digit (for example, rounding to 1/10 can be applied after rounding to 1/100).
For example, when rounding to integer,
* 20.0 is rounded to 20;
* 20.01, 20.1, 20.9, 20.99, 21, 21.01, 21.9, 21.99 are rounded to 21 (avoiding a final 0);
* 22.0, 22.1, 22.9, 22.99 are rounded to 22;
* 24.0, 24.1, 24.9, 24.99 are rounded to 24 (avoiding a final 5);
* 25.0 is rounded to 25;
* 25.01, 25.1 are rounded to 26 (avoiding a final 5).

In the example from "[[#Double rounding|Double rounding]]" section, rounding 9.46 to one decimal gives 9.4, which rounding to integer in turn gives 9.

With binary arithmetic, this rounding is also called "round to odd" (not to be confused with "[[#Rounding half to odd|round half to odd]]"). For example, when rounding to 1/4 (0.01 in binary),
* {{math|1=''x'' = 2.0}} ⇒ result is 2 (10.00 in binary)
* {{math|2.0 < ''x'' < 2.5}} ⇒ result is 2.25 (10.01 in binary)
* {{math|1=''x'' = 2.5}} ⇒ result is 2.5 (10.10 in binary)
* {{math|2.5 < ''x'' < 3.0}} ⇒ result is 2.75 (10.11 in binary)
* {{math|1=''x'' = 3.0}} ⇒ result is 3 (11.00 in binary)

For correct results with binary arithmetic, each rounding step must remove at least 2 binary digits, otherwise, wrong results may appear. For example,
* 3.125 RPSP to 1/4 ⇒ result is 3.25
* 3.25 RPSP to 1/2 ⇒ result is 3.5
* 3.5 round-half-to-even to 1 ⇒ result is 4 (wrong)

If the erroneous middle step is removed, the final rounding to integer rounds 3.25 to the correct value of 3.

RPSP is implemented in hardware in IBM [[zSeries]] and [[pSeries]]. In [[Python (programming language)|Python]] module "Decimal", [[Tcl]] module "math", [[Haskell]] package "decimal-arithmetic", and possibly others, this mode is called ROUND_05UP or round05up.

===Randomized rounding to an integer===

====Alternating tie-breaking====
One method, more obscure than most, is to alternate direction when rounding a number with 0.5 fractional part. All others are rounded to the closest integer. Whenever the fractional part is 0.5, alternate rounding up or down: for the first occurrence of a 0.5 fractional part, round up, for the second occurrence, round down, and so on. Alternatively, the first 0.5 fractional part rounding can be determined by a [[random seed]].  "Up" and "down" can be any two rounding methods that oppose each other - toward and away from positive infinity or toward and away from zero.

If occurrences of 0.5 fractional parts occur significantly more than a restart of the occurrence "counting", then it is effectively bias free. With guaranteed zero bias, it is useful if the numbers are to be summed or averaged.

====Random tie-breaking====
If the fractional part of {{mvar|x}} is 0.5, choose {{mvar|y}} randomly between {{math|''x'' + 0.5}} and {{math|''x'' − 0.5}}, with equal probability. All others are rounded to the closest integer.

Like round-half-to-even and round-half-to-odd, this rule is essentially free of overall bias, but it is also fair among even and odd {{mvar|y}} values. An advantage over alternate tie-breaking is that the last direction of rounding on the 0.5 fractional part does not have to be "remembered".

====Stochastic rounding====
Rounding as follows to one of the closest integer toward negative infinity and the closest integer toward positive infinity, with a probability dependent on the proximity is called [[stochastic]] rounding and will give an unbiased result on average.<ref name="stochastic">{{cite arXiv |title=Deep Learning with Limited Numerical Precision |eprint=1502.02551 |first1=Suyog |last1=Gupta |first2=Ankur |last2=Angrawl |first3=Kailash |last3=Gopalakrishnan |first4=Pritish |last4=Narayanan |page=3 |date=9 February 2016 |class=cs.LG}}</ref>
:<math>\operatorname {Round} (x) = \begin{cases}
\lfloor x \rfloor & \text { with probability } 1 - (x - \lfloor x \rfloor) = \lfloor x \rfloor - x + 1 \\[5mu]
\lfloor x \rfloor + 1 & \text { with probability } {x - \lfloor x \rfloor}
\end{cases}
</math>

For example, 1.6 would be rounded to 1 with probability 0.4 and to 2 with probability 0.6.

Stochastic rounding can be accurate in a way that a rounding [[function (mathematics)|function]] can never be. For example, suppose one started with 0 and added 0.3 to that one hundred times while rounding the running total between every addition. The result would be 0 with regular rounding, but with stochastic rounding, the expected result would be 30, which is the same value obtained without rounding. This can be useful in [[machine learning]] where the training may use low precision arithmetic iteratively.<ref name="stochastic"/> Stochastic rounding is also a way to achieve 1-dimensional [[dithering]].

===Comparison of approaches for rounding to an integer===
{|class="wikitable" style="text-align:center; border-width:0;"
|-
! rowspan="3" | Value
! colspan="11"| Functional methods
! colspan="6" | Randomized methods
|-
! colspan="4" | Directed rounding
! colspan="6" | Round to nearest
! rowspan="2" | Round to prepare for shorter precision
! colspan="2" | Alternating tie-break
! colspan="2" | Random tie-break
! colspan="2" | Stochastic
|- style="line-height:110%;"
! Down<br /><small>(toward −<big><big>∞</big></big>)</small>
! Up<br /><small>(toward +<big><big>∞</big></big>)</small>
! Toward 0
! Away From 0
! Half Down<br /><small>(toward −<big><big>∞</big></big>)</small>
! Half Up<br /><small>(toward +<big><big>∞</big></big>)</small>
! Half Toward 0
! Half Away From 0
! Half to Even
! Half to Odd
![[Arithmetic mean|Average]]
![[Standard deviation|SD]]
![[Arithmetic mean|Average]]
![[Standard deviation|SD]]
![[Arithmetic mean|Average]]
![[Standard deviation|SD]]
|-
| +2.8
| rowspan="3" | +2
| rowspan="3" | +3
| rowspan="3" | +2
| rowspan="3" | +3
| +3
| rowspan="2" | +3
| +3
| rowspan="2" | +3
| +3
| rowspan="2" | +3
| rowspan="3" | +2
| +3
|0
| +3
|0
| +2.8
|0.04
|-
| +2.5
| rowspan="3" | +2
| rowspan="3" | +2
| rowspan="4" | +2
| +2.505
|0
| +2.5
|0.05
| +2.5
|0.05
|-
| +2.2
| rowspan="3" | +2
| rowspan="3" | +2
| rowspan="2" | +2
| rowspan="2" | +2
| rowspan="2" |0
| rowspan="2" | +2
| rowspan="2" |0
| +2.2
|0.04
|-
| +1.8
| rowspan="3" | +1
| rowspan="3" | +2
| rowspan="3" | +1
| rowspan="3" | +2
| rowspan="6" | +1
| +1.8
| 0.04
|-
| +1.5
| rowspan="3" | +1
| rowspan="3" | +1
| rowspan="4" | +1
| +1.505
| 0
| +1.5
| 0.05
| +1.5
| 0.05
|-
| +1.2
| rowspan="3" | +1
| rowspan="3" | +1
| rowspan="2" | +1
| rowspan="2" | +1
| rowspan="2" | 0
| rowspan="2" | +1
| rowspan="2" | 0
| +1.2
| 0.04
|-
| +0.8
| rowspan="3" | 0
| rowspan="3" | +1
| rowspan="6" | 0
| rowspan="3" | +1
| +0.8
| 0.04
|-
| +0.5
| rowspan="3" | 0
| rowspan="4" | 0
| rowspan="4" | 0
| +0.505
| 0
| +0.5
| 0.05
| +0.5
| 0.05
|-
| +0.2
| rowspan="3" | 0
| rowspan="2" | 0
| rowspan="2" | 0
| rowspan="2" | 0
| rowspan="2" | 0
| rowspan="2" | 0
| rowspan="2" | 0
| +0.2
| 0.04
|-
| −0.2
| rowspan="3" | −1
| rowspan="3" | 0
| rowspan="3" | −1
| rowspan="6" | −1
|−0.2
| 0.04
|-
| −0.5
| rowspan="3" | −1
| rowspan="3" | −1
| rowspan="4" | −1
|−0.495
| 0
|−0.5
| 0.05
|−0.5
| 0.05
|-
| −0.8
| rowspan="3" | −1
| rowspan="3" | −1
| rowspan="2" | −1
| rowspan="2" | −1
| rowspan="2" | 0
| rowspan="2" | −1
| rowspan="2" | 0
|−0.8
| 0.04
|-
| −1.2
| rowspan="3" | −2
| rowspan="3" | −1
| rowspan="3" | −1
| rowspan="3" | −2
|−1.2
| 0.04
|-
| −1.5
| rowspan="3" | −2
| rowspan="3" | −2
| rowspan="4" | −2
|−1.495
| 0
|−1.5
| 0.05
|−1.5
| 0.05
|-
| −1.8
| rowspan="3" | −2
| rowspan="3" | -2
| rowspan="2" | −2
| rowspan="2" | −2
| rowspan="2" | 0
| rowspan="2" | −2
| rowspan="2" | 0
|−1.8
| 0.04
|-
|−2.2
| rowspan="3" |−3
| rowspan="3" |−2
| rowspan="3" |−2
| rowspan="3" |−3
| rowspan="3" |−2
|−2.2
|0.04
|-
|−2.5
| rowspan="2" |−3
| rowspan="2" |−3
| rowspan="2" |−3
|−2.495
|0
|−2.5
|0.05
|−2.5
|0.05
|-
|−2.8
|−3
|−3
|−3
|−3
|0
|−3
|0
|−2.8
|0.04
|}