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Fixing some issues on text and removing redundant subscripts from ope…
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…rations of some equations
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caiolima committed Apr 16, 2019
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Expand Up @@ -870,21 +870,21 @@ <h1>Mathematical Operations</h1>
<li><em>Mathematical value</em>: Arbitrary real numbers, used for specific situations.</li>
</ul>

<p>In the language of this specification, numerical values and operations (including addition, subtraction, negation, multiplication, addition, and comparison) are distinguished among different numeric kinds using subscripts. The subscript <sub><dfn id="𝔽">𝔽</dfn></sub> refers to Numbers, and the subscript <sub><dfn id="ℝ">ℝ</dfn></sub> refers to mathematical values. A subscript is used following each numeric value and operation.</p>
<p>In the language of this specification, numerical values and operations (including addition, subtraction, negation, multiplication, division, and comparison) are distinguished among different numeric kinds using subscripts. The subscript <sub><dfn id="𝔽">𝔽</dfn></sub> refers to Numbers, and the subscript <sub><dfn id="ℝ">ℝ</dfn></sub> refers to mathematical values. A subscript is used following each numeric value and operation.</p>
<p>For brevity, the <sub>𝔽</sub> subscript can be omitted on Number values--a numeric value with no subscript is interpreted to be a Number. An operation with no subscript is interpreted to be a Number operation, unless one of the parameters has a particular subscript, in which case the operation adopts that subscript. For example, 1<sub>ℝ</sub> + 2<sub>ℝ</sub> = 3<sub>ℝ</sub> is a statement about mathematical values, and 1 + 2 = 3 is a statement about Numbers.</p>
<p>In general, when this specification refers to a numerical value, such as in the phrase, "the length of _y_" or "the integer represented by the four hexadecimal digits ...", without explicitly specifying a numeric kind, the phrase refers to a Number. Phrases which refer to a mathematical value are explicitly annotated as such; for example, "the mathematical value of the number of code points in ...".</p>
<p>It is not defined to mix Numbers and mathematical values in either arithmetic or comparison operations, and any such undefined operation would be an editorial error in this specification text.</p>
<p>The Number value 0, alternatively written 0<sub>𝔽</sub>, is defined as the double-precision floating point positive zero value. In certain contexts, it may also be written as +0 for clarity.</p>
<p>This specification denotes most numeric values in base 10; it also uses numeric values of the form 0x followed by digits 0-9 or A-F as base-16 values.</p>
<p>In certain contexts, an operation is specified which is generic between Numbers and mathematical values. In these cases, the subscript can be a variable; _t_ is often used for this purpose, for example 5<sub>_t_</sub> &times; 10<sub>_t_</sub> =<sub>_t_</sub> 50<sub>_t_</sub> for any _t_ ranging over ℝ and 𝔽, since the values involved are within the range where the semantics coincide.</p>
<p>In certain contexts, an operation is specified which is generic between Numbers and mathematical values. In these cases, the subscript can be a variable; _t_ is often used for this purpose, for example 5<sub>_t_</sub> &times; 10<sub>_t_</sub> = 50<sub>_t_</sub> for any _t_ ranging over ℝ and 𝔽, since the values involved are within the range where the semantics coincide.</p>
<p>Conversions between mathematical values and numbers are never implicit, and always explicit in this document. A conversion from a mathematical value to a Number is denoted as "the Number value for _x_", and is defined in <emu-xref href="#sec-ecmascript-language-types-number-type"></emu-xref>. A conversion from a Number to a mathematical value is denoted as "the <dfn id="mathematical-value">mathematical value</dfn> of _x_", or ℝ(_x_). Note that the mathematical value of non-finite values is not defined, and the mathematical value of *+0* and *-0* is the mathematical value 0<sub>ℝ</sub>.</p>
<p>When the term <dfn id="integer">integer</dfn> is used in this specification, it refers to a Number value whose mathematical value is in the set of integers, unless otherwise stated: when the term <dfn id="mathematical integer">mathematical integer</dfn> is used in this specification, it refers to a mathematical value which is in the set of integers. As shorthand, integer<sub>_t_</sub> can be used to refer to either of the two, as determined by _t_.</p>
<p>The mathematical function <emu-eqn id="eqn-abs" aoid="abs">abs<sub>_t_</sub>(_x_)</emu-eqn> produces the absolute value of _x_, which is <emu-eqn>-<sub>_t_</sub>_x_</emu-eqn> if _x_ &lt;<sub>_t_</sub> 0<sub>_t_</sub> and otherwise is _x_ itself.</p>
<p>The mathematical function <emu-eqn id="eqn-min" aoid="min">min<sub>_t_</sub>(_x1_, _x2_, ..., _xN_)</emu-eqn> produces the mathematically smallest of <emu-eqn>_x1_</emu-eqn> through <emu-eqn>_xN_</emu-eqn>. The mathematical function <emu-eqn id="eqn-max" aoid="max">max<sub>_t_</sub>(_x1_, _x2_, ..., _xN_)</emu-eqn> produces the mathematically largest of <emu-eqn>_x1_</emu-eqn> through <emu-eqn>_xN_</emu-eqn>. The domain and range of these mathematical functions include *+&infin;* and *-&infin;*.</p>
<p>The notation &ldquo;<emu-eqn id="eqn-modulo" aoid="modulo">_x_ modulo<sub>_t_</sub> _y_</emu-eqn>&rdquo; (_y_ must be finite and nonzero) computes a value _k_ of the same sign as _y_ (or zero) such that <emu-eqn>abs<sub>_t_</sub>(_k_) &lt;<sub>_t_</sub> abs<sub>_t_</sub>(_y_) and _x_-<sub>_t_</sub>_k_ = _q_ &times;<sub>_t_</sub> _y_</emu-eqn> for some integer<sub>_t_</sub> _q_.</p>
<p>The mathematical function <emu-eqn id="eqn-floor" aoid="floor">floor<sub>_t_</sub>(_x_)</emu-eqn> produces the largest integer<sub>_t_</sub> (closest to positive infinity) that is not larger than _x_.</p>
<emu-note>
<p><emu-eqn>floor<sub>_t_</sub>(_x_) = _x_-(_x_ modulo<sub>_t_</sub> 1<sub>_t_</sub>)</emu-eqn>.</p>
<p><emu-eqn>floor<sub>_t_</sub>(_x_) = _x_ -<sub>_t_</sub> (_x_ modulo<sub>_t_</sub> 1<sub>_t_</sub>)</emu-eqn>.</p>
</emu-note>
</emu-clause>
</emu-clause>
Expand Down Expand Up @@ -3777,13 +3777,13 @@ <h1>Runtime Semantics: MV</h1>
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: DecimalDigits `.` ExponentPart</emu-grammar> is the MV of |DecimalDigits| times 10<sub>ℝ</sub><sup>_e_</sup>, where _e_ is the MV of |ExponentPart|.
</li>
<li>
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: DecimalDigits `.` DecimalDigits ExponentPart</emu-grammar> is (the MV of the first |DecimalDigits| plus (the MV of the second |DecimalDigits| &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>-<sub>ℝ</sub>_n_</sup>)) times 10<sub>ℝ</sub><sup>_e_</sup>, where _n_ is the mathematical value of the number of code points in the second |DecimalDigits| and _e_ is the MV of |ExponentPart|.
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: DecimalDigits `.` DecimalDigits ExponentPart</emu-grammar> is (the MV of the first |DecimalDigits| plus (the MV of the second |DecimalDigits| times 10<sub>ℝ</sub><sup>-<sub>ℝ</sub>_n_</sup>)) times 10<sub>ℝ</sub><sup>_e_</sup>, where _n_ is the mathematical value of the number of code points in the second |DecimalDigits| and _e_ is the MV of |ExponentPart|.
</li>
<li>
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: `.` DecimalDigits</emu-grammar> is the MV of |DecimalDigits| times 10<sub>ℝ</sub><sup>-<sub>ℝ</sub>_n_</sup>, where _n_ is the mathematical value of the number of code points in |DecimalDigits|.
</li>
<li>
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: `.` DecimalDigits ExponentPart</emu-grammar> is the MV of |DecimalDigits| times 10<sub>ℝ</sub><sup>_e_-<sub>ℝ</sub>_n_</sup>, where _n_ is the mathematical value of the number of code points in |DecimalDigits| and _e_ is the MV of |ExponentPart|.
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: `.` DecimalDigits ExponentPart</emu-grammar> is the MV of |DecimalDigits| times 10<sub>ℝ</sub><sup>_e_ -<sub>ℝ</sub> _n_</sup>, where _n_ is the mathematical value of the number of code points in |DecimalDigits| and _e_ is the MV of |ExponentPart|.
</li>
<li>
The MV of <emu-grammar>StrUnsignedDecimalLiteral ::: DecimalDigits</emu-grammar> is the MV of |DecimalDigits|.
Expand Down Expand Up @@ -4036,7 +4036,7 @@ <h1>NumberToString ( _m_ )</h1>
1. If _m_ is *+0* or *-0*, return the String `"0"`.
1. If _m_ is less than zero, return the string-concatenation of `"-"` and ! NumberToString(-_m_).
1. If _m_ is *+&infin;*, return the String `"Infinity"`.
1. Otherwise, let _n_, _k_, and _s_ be integers such that _k_ &ge; 1, 10<sup>_k_ - 1</sup> &le; _s_ &lt; 10<sup>_k_</sup>, the Number value for ℝ(_s_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_n_) -<sub>ℝ</sub> ℝ(_k_)</sup> is _m_, and _k_ is as small as possible. Note that _k_ is the number of digits in the decimal representation of _s_, that _s_ is not divisible by 10<sub>ℝ</sub>, and that the least significant digit of _s_ is not necessarily uniquely determined by these criteria.
1. Otherwise, let _n_, _k_, and _s_ be integers such that _k_ &ge; 1, 10<sup>_k_ - 1</sup> &le; _s_ &lt; 10<sup>_k_</sup>, the Number value for ℝ(_s_) &times; 10<sub>ℝ</sub><sup>ℝ(_n_) - ℝ(_k_)</sup> is _m_, and _k_ is as small as possible. Note that _k_ is the number of digits in the decimal representation of _s_, that _s_ is not divisible by 10<sub>ℝ</sub>, and that the least significant digit of _s_ is not necessarily uniquely determined by these criteria.
1. If _k_ &le; _n_ &le; 21, return the string-concatenation of:
* the code units of the _k_ digits of the decimal representation of _s_ (in order, with no leading zeroes)
* _n_ - _k_ occurrences of the code unit 0x0030 (DIGIT ZERO)
Expand Down Expand Up @@ -4076,7 +4076,7 @@ <h1>NumberToString ( _m_ )</h1>
<emu-note>
<p>For implementations that provide more accurate conversions than required by the rules above, it is recommended that the following alternative version of step 5 be used as a guideline:</p>
<emu-alg>
5. Otherwise, let _n_, _k_, and _s_ be integers such that _k_ &ge; 1, 10<sup>_k_ - 1</sup> &le; _s_ &lt; 10<sup>_k_</sup>, the Number value for ℝ(_s_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_n_) -<sub>ℝ</sub> ℝ(_k_)</sup> is _m_, and _k_ is as small as possible. If there are multiple possibilities for _s_, choose the value of _s_ for which ℝ(_s_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_n_) -<sub>ℝ</sub> ℝ(_k_)</sup> is closest in value to ℝ(_m_). If there are two such possible values of _s_, choose the one that is even. Note that _k_ is the number of digits in the decimal representation of _s_ and that _s_ is not divisible by 10<sub>ℝ</sub>.
5. Otherwise, let _n_, _k_, and _s_ be integers such that _k_ &ge; 1, 10<sup>_k_ - 1</sup> &le; _s_ &lt; 10<sup>_k_</sup>, the Number value for ℝ(_s_) &times; 10<sub>ℝ</sub><sup>ℝ(_n_) - ℝ(_k_)</sup> is _m_, and _k_ is as small as possible. If there are multiple possibilities for _s_, choose the value of _s_ for which ℝ(_s_) &times; 10<sub>ℝ</sub><sup>ℝ(_n_) - ℝ(_k_)</sup> is closest in value to ℝ(_m_). If there are two such possible values of _s_, choose the one that is even. Note that _k_ is the number of digits in the decimal representation of _s_ and that _s_ is not divisible by 10<sub>ℝ</sub>.
</emu-alg>
</emu-note>
<emu-note>
Expand Down Expand Up @@ -25743,9 +25743,9 @@ <h1>Number.prototype.toExponential ( _fractionDigits_ )</h1>
1. Let _e_ be 0.
1. Else _x_ &ne; 0,
1. If _fractionDigits_ is not *undefined*, then
1. Let _e_ and _n_ be integers such that 10<sup>_f_</sup> &le; _n_ &lt; 10<sup>_f_ + 1</sup> and for which ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_) -<sub>ℝ</sub> ℝ(_n_)</sup> - ℝ(_x_) is as close to zero as possible. If there are two such sets of _e_ and _n_, pick the _e_ and _n_ for which ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_) -<sub>ℝ</sub> ℝ(_f_)</sup> is larger.
1. Let _e_ and _n_ be integers such that 10<sup>_f_</sup> &le; _n_ &lt; 10<sup>_f_ + 1</sup> and for which ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_n_)</sup> - ℝ(_x_) is as close to zero as possible. If there are two such sets of _e_ and _n_, pick the _e_ and _n_ for which ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_f_)</sup> is larger.
1. Else _fractionDigits_ is *undefined*,
1. Let _e_, _n_, and _f_ be integers such that _f_ &ge; 0, 10<sup>_f_</sup> &le; _n_ &lt; 10<sup>_f_ + 1</sup>, the Number value for ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_f_)</sup> is _x_, and _f_ is as small as possible. Note that the decimal representation of _n_ has _f_ + 1<sub>ℝ</sub> digits, _n_ is not divisible by 10, and the least significant digit of _n_ is not necessarily uniquely determined by these criteria.
1. Let _e_, _n_, and _f_ be integers such that _f_ &ge; 0, 10<sup>_f_</sup> &le; _n_ &lt; 10<sup>_f_ + 1</sup>, the Number value for ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_f_)</sup> is _x_, and _f_ is as small as possible. Note that the decimal representation of _n_ has _f_ + 1<sub>ℝ</sub> digits, _n_ is not divisible by 10, and the least significant digit of _n_ is not necessarily uniquely determined by these criteria.
1. Let _m_ be the String value consisting of the digits of the decimal representation of _n_ (in order, with no leading zeroes).
1. If _f_ &ne; 0, then
1. Let _a_ be the first code unit of _m_, and let _b_ be the remaining _f_ code units of _m_.
Expand All @@ -25765,7 +25765,7 @@ <h1>Number.prototype.toExponential ( _fractionDigits_ )</h1>
<emu-note>
<p>For implementations that provide more accurate conversions than required by the rules above, it is recommended that the following alternative version of step 10.b.i be used as a guideline:</p>
<emu-alg type="i">
1. Let _e_, _n_, and _f_ be integers such that _f_ &ge; 0, 10<sup>_f_</sup> &le; _n_ &lt; 10<sup>_f_+1</sup>, the Number value for ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_)-<sub>ℝ</sub>ℝ(_f_)</sup> is _x_, and _f_ is as small as possible. If there are multiple possibilities for _n_, choose the value of _n_ for which ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_)-<sub>ℝ</sub>ℝ(_f_)</sup> is closest in value to _x_. If there are two such possible values of _n_, choose the one that is even.
1. Let _e_, _n_, and _f_ be integers such that _f_ &ge; 0, 10<sup>_f_</sup> &le; _n_ &lt; 10<sup>_f_ + 1</sup>, the Number value for ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_f_)</sup> is _x_, and _f_ is as small as possible. If there are multiple possibilities for _n_, choose the value of _n_ for which ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_f_)</sup> is closest in value to _x_. If there are two such possible values of _n_, choose the one that is even.
</emu-alg>
</emu-note>
</emu-clause>
Expand All @@ -25789,7 +25789,7 @@ <h1>Number.prototype.toFixed ( _fractionDigits_ )</h1>
1. If _x_ &ge; 10<sup>21</sup>, then
1. Let _m_ be ! ToString(_x_).
1. Else _x_ &lt; 10<sup>21</sup>,
1. Let _n_ be an integer for which ℝ(_n_) &divide;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_f_)</sup> - ℝ(_x_) is as close to zero as possible. If there are two such _n_, pick the larger _n_.
1. Let _n_ be an integer for which ℝ(_n_) &divide; 10<sub>ℝ</sub><sup>ℝ(_f_)</sup> - ℝ(_x_) is as close to zero as possible. If there are two such _n_, pick the larger _n_.
1. If _n_ = 0, let _m_ be the String `"0"`. Otherwise, let _m_ be the String value consisting of the digits of the decimal representation of _n_ (in order, with no leading zeroes).
1. If _f_ &ne; 0, then
1. Let _k_ be the length of _m_.
Expand Down Expand Up @@ -25835,7 +25835,7 @@ <h1>Number.prototype.toPrecision ( _precision_ )</h1>
1. Let _m_ be the String value consisting of _p_ occurrences of the code unit 0x0030 (DIGIT ZERO).
1. Let _e_ be 0.
1. Else _x_ &ne; 0,
1. Let _e_ and _n_ be integers such that 10<sup>_p_ - 1</sup> &le; _n_ &lt; 10<sup>_p_</sup> and for which ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_p_) + 1<sub>ℝ</sub></sup> - ℝ(_x_) is as close to zero as possible. If there are two such sets of _e_ and _n_, pick the _e_ and _n_ for which ℝ(_n_) &times;<sub>ℝ</sub> 10<sub>ℝ</sub><sup>ℝ(_e_) -<sub>ℝ</sub> ℝ(_p_) + 1<sub>ℝ</sub></sup> is larger.
1. Let _e_ and _n_ be integers such that 10<sup>_p_ - 1</sup> &le; _n_ &lt; 10<sup>_p_</sup> and for which ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_p_) + 1<sub>ℝ</sub></sup> - ℝ(_x_) is as close to zero as possible. If there are two such sets of _e_ and _n_, pick the _e_ and _n_ for which ℝ(_n_) &times; 10<sub>ℝ</sub><sup>ℝ(_e_) - ℝ(_p_) + 1<sub>ℝ</sub></sup> is larger.
1. Let _m_ be the String value consisting of the digits of the decimal representation of _n_ (in order, with no leading zeroes).
1. If _e_ &lt; -6 or _e_ &ge; _p_, then
1. Assert: _e_ &ne; 0.
Expand Down

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