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Editorial: Introduce term "non-zero finite Number" #3917

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Editorial: Introduce term "non-zero finite Number" #3917

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39 changes: 16 additions & 23 deletions spec.html
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Expand Up @@ -1963,17 +1963,11 @@ <h1>The Number Type</h1>
<p>There are two other special values, called *positive Infinity* and *negative Infinity*. For brevity, these values are also referred to for expository purposes by the symbols *+∞*<sub>𝔽</sub> and *-∞*<sub>𝔽</sub>, respectively. (Note that these two infinite Number values are produced by the program expressions `+Infinity` (or simply `Infinity`) and `-Infinity`.)</p>
<p>Each of the other 18,437,736,874,454,810,624 (that is, <emu-eqn>2<sup>64</sup> - 2<sup>53</sup></emu-eqn>) values is a <dfn id="finite-number" variants="finite Numbers" oldids="finite">finite Number</dfn>. Half of these are positive numbers and half are negative numbers; for every positive finite Number, there is a corresponding negative Number having the same magnitude.</p>
<p>Note that there is both a *positive zero* and a *negative zero*. For brevity, these values are also referred to for expository purposes by the symbols *+0*<sub>𝔽</sub> and *-0*<sub>𝔽</sub>, respectively. (Note that these two different zero Number values are produced by the program expressions `+0` (or simply `0`) and `-0`.)</p>
<p>The 18,437,736,874,454,810,622 (that is, <emu-eqn>2<sup>64</sup> - 2<sup>53</sup> - 2</emu-eqn>) finite non-zero values are of two kinds:</p>
<p>18,428,729,675,200,069,632 (that is, <emu-eqn>2<sup>64</sup> - 2<sup>54</sup></emu-eqn>) of them are normalized, having the form</p>
<div class="math-display">
_s_ × _m_ × 2<sup>_e_</sup>
</div>
<p>where _s_ is 1 or -1, _m_ is an integer in the interval from 2<sup>52</sup> (inclusive) to 2<sup>53</sup> (exclusive), and _e_ is an integer in the inclusive interval from -1074 to 971.</p>
<p>The remaining 9,007,199,254,740,990 (that is, <emu-eqn>2<sup>53</sup> - 2</emu-eqn>) values are denormalized, having the form</p>
<div class="math-display">
_s_ × _m_ × 2<sup>_e_</sup>
</div>
<p>where _s_ is 1 or -1, _m_ is an integer in the interval from 0 (exclusive) to 2<sup>52</sup> (exclusive), and _e_ is -1074.</p>
<p>The 18,437,736,874,454,810,622 (that is, <emu-eqn>2<sup>64</sup> - 2<sup>53</sup> - 2</emu-eqn>) <dfn id="non-zero-finite-number" variants="non-zero finite Number,finite non-zero Number,finite non-zero Numbers">non-zero finite Numbers</dfn> are partitioned into two groups, based on integers _m_ and _e_ in their corresponding mathematical value <emu-eqn>_s_ × _m_ × 2<sup>_e_</sup></emu-eqn>, where _s_ is 1 or -1:</p>
<ul>
<li>18,428,729,675,200,069,632 (that is, <emu-eqn>2<sup>64</sup> - 2<sup>54</sup></emu-eqn>) of them are “normalized”: _m_ is an integer in the interval from 2<sup>52</sup> (inclusive) to 2<sup>53</sup> (exclusive), and _e_ is an integer in the inclusive interval from -1074 to 971.</li>
<li>The remaining 9,007,199,254,740,990 (that is, <emu-eqn>2<sup>53</sup> - 2</emu-eqn>) are “denormalized” and more specifically “subnormal”: _m_ is an integer in the interval from 0 (exclusive) to 2<sup>52</sup> (exclusive), and _e_ is -1074.</li>
</ul>
<p>Note that all the positive and negative integers whose magnitude is no greater than 2<sup>53</sup> are representable in the Number type. The integer 0 has two representations in the Number type: *+0*<sub>𝔽</sub> and *-0*<sub>𝔽</sub>.</p>
<p>A finite Number has an <em>odd significand</em> if it is non-zero and the integer _m_ used to express it (in one of the two forms shown above) is odd. Otherwise, it has an <em>even significand</em>.</p>
<p>In this specification, the phrase “the <dfn id="number-value-for" oldids="number-value">Number value for</dfn> _x_” where _x_ represents an exact real mathematical quantity (which might even be an irrational number such as π) means a Number value chosen in the following manner. Consider the set consisting of the union of all finite Numbers except *-0*<sub>𝔽</sub> with two additional values that are not representable in the Number type, namely 2<sup>1024</sup> (which is <emu-eqn>+1 × 2<sup>53</sup> × 2<sup>971</sup></emu-eqn>) and <emu-eqn>-2<sup>1024</sup></emu-eqn> (which is <emu-eqn>-1 × 2<sup>53</sup> × 2<sup>971</sup></emu-eqn>). Choose the member of this set that is closest in value to _x_. If two values of the set are equally close, then the one with an even significand is chosen; for this purpose, the two extra values 2<sup>1024</sup> and <emu-eqn>-2<sup>1024</sup></emu-eqn> are considered to have even significands. Finally, if 2<sup>1024</sup> was chosen, replace it with *+∞*<sub>𝔽</sub>; if <emu-eqn>-2<sup>1024</sup></emu-eqn> was chosen, replace it with *-∞*<sub>𝔽</sub>; if *+0*<sub>𝔽</sub> was chosen, replace it with *-0*<sub>𝔽</sub> if and only if _x_ &lt; 0; any other chosen value is used unchanged. The result is the Number value for _x_. (This procedure corresponds exactly to the behaviour of the IEEE 754-2019 roundTiesToEven mode.)</p>
Expand Down Expand Up @@ -2041,7 +2035,7 @@ <h1>
1. Return *+0*<sub>𝔽</sub>.
1. If _exponent_ is an odd integral Number, return *-∞*<sub>𝔽</sub>.
1. Return *+∞*<sub>𝔽</sub>.
1. Assert: _base_ is a finite Number and is neither *+0*<sub>𝔽</sub> nor *-0*<sub>𝔽</sub>.
1. Assert: _base_ is a non-zero finite Number.
1. If _exponent_ is *+∞*<sub>𝔽</sub>, then
1. If abs(ℝ(_base_)) > 1, return *+∞*<sub>𝔽</sub>.
1. If abs(ℝ(_base_)) = 1, return *NaN*.
Expand All @@ -2050,7 +2044,7 @@ <h1>
1. If abs(ℝ(_base_)) > 1, return *+0*<sub>𝔽</sub>.
1. If abs(ℝ(_base_)) = 1, return *NaN*.
1. Return *+∞*<sub>𝔽</sub>.
1. Assert: _exponent_ is a finite Number and is neither *+0*<sub>𝔽</sub> nor *-0*<sub>𝔽</sub>.
1. Assert: _exponent_ is a non-zero finite Number.
1. If _base_ &lt; *-0*<sub>𝔽</sub> and _exponent_ is not an integral Number, return *NaN*.
1. Return an implementation-approximated Number value representing the result of raising ℝ(_base_) to the ℝ(_exponent_) power.
</emu-alg>
Expand Down Expand Up @@ -2147,8 +2141,7 @@ <h1>
1. If _denominator_ is either *+∞*<sub>𝔽</sub> or *-∞*<sub>𝔽</sub>, return _numerator_.
1. If _denominator_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return *NaN*.
1. If _numerator_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _numerator_.
1. Assert: _numerator_ is a finite Number and is neither *+0*<sub>𝔽</sub> nor *-0*<sub>𝔽</sub>.
1. Assert: _denominator_ is a finite Number and is neither *+0*<sub>𝔽</sub> nor *-0*<sub>𝔽</sub>.
1. Assert: _numerator_ and _denominator_ are non-zero finite Numbers.
1. Let _quotient_ be ℝ(_numerator_) / ℝ(_denominator_).
1. Let _truncatedQuotient_ be truncate(_quotient_).
1. Let _remainder_ be ℝ(_numerator_) - (ℝ(_denominator_) × _truncatedQuotient_).
Expand Down Expand Up @@ -33463,7 +33456,7 @@ <h1>Math.asinh ( _x_ )</h1>
<p>It performs the following steps when called:</p>
<emu-alg>
1. Let _n_ be ? ToNumber(_x_).
1. If _n_ is not a finite Number or _n_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _n_.
1. If _n_ is not a non-zero finite Number, return _n_.
1. Return an implementation-approximated Number value representing the inverse hyperbolic sine of ℝ(_n_).
</emu-alg>
</emu-clause>
Expand Down Expand Up @@ -33517,7 +33510,7 @@ <h1>Math.atan2 ( _y_, _x_ )</h1>
1. If _ny_ is *-0*<sub>𝔽</sub>, then
1. If _nx_ > *+0*<sub>𝔽</sub> or _nx_ is *+0*<sub>𝔽</sub>, return *-0*<sub>𝔽</sub>.
1. Return an implementation-approximated Number value representing -π.
1. Assert: _ny_ is a finite Number and is neither *+0*<sub>𝔽</sub> nor *-0*<sub>𝔽</sub>.
1. Assert: _ny_ is a non-zero finite Number.
1. If _ny_ > *+0*<sub>𝔽</sub>, then
1. If _nx_ is *+∞*<sub>𝔽</sub>, return *+0*<sub>𝔽</sub>.
1. If _nx_ is *-∞*<sub>𝔽</sub>, return an implementation-approximated Number value representing π.
Expand All @@ -33526,7 +33519,7 @@ <h1>Math.atan2 ( _y_, _x_ )</h1>
1. If _nx_ is *+∞*<sub>𝔽</sub>, return *-0*<sub>𝔽</sub>.
1. If _nx_ is *-∞*<sub>𝔽</sub>, return an implementation-approximated Number value representing -π.
1. If _nx_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return an implementation-approximated Number value representing -π / 2.
1. Assert: _nx_ is a finite Number and is neither *+0*<sub>𝔽</sub> nor *-0*<sub>𝔽</sub>.
1. Assert: _nx_ is a non-zero finite Number.
1. Let _result_ be the inverse tangent of abs(ℝ(_ny_) / ℝ(_nx_)).
1. If _nx_ &lt; *-0*<sub>𝔽</sub>, then
1. If _ny_ > *+0*<sub>𝔽</sub>, set _result_ to π - _result_.
Expand All @@ -33543,7 +33536,7 @@ <h1>Math.cbrt ( _x_ )</h1>
<p>It performs the following steps when called:</p>
<emu-alg>
1. Let _n_ be ? ToNumber(_x_).
1. If _n_ is not a finite Number or _n_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _n_.
1. If _n_ is not a non-zero finite Number, return _n_.
1. Return an implementation-approximated Number value representing the cube root of ℝ(_n_).
</emu-alg>
</emu-clause>
Expand All @@ -33554,7 +33547,7 @@ <h1>Math.ceil ( _x_ )</h1>
<p>It performs the following steps when called:</p>
<emu-alg>
1. Let _n_ be ? ToNumber(_x_).
1. If _n_ is not a finite Number or _n_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _n_.
1. If _n_ is not a non-zero finite Number, return _n_.
1. If _n_ &lt; *-0*<sub>𝔽</sub> and _n_ > *-1*<sub>𝔽</sub>, return *-0*<sub>𝔽</sub>.
1. If _n_ is an integral Number, return _n_.
1. Return the smallest (closest to -∞) integral Number value that is not less than _n_.
Expand Down Expand Up @@ -33637,7 +33630,7 @@ <h1>Math.floor ( _x_ )</h1>
<p>It performs the following steps when called:</p>
<emu-alg>
1. Let _n_ be ? ToNumber(_x_).
1. If _n_ is not a finite Number or _n_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _n_.
1. If _n_ is not a non-zero finite Number, return _n_.
1. If _n_ &lt; *1*<sub>𝔽</sub> and _n_ > *+0*<sub>𝔽</sub>, return *+0*<sub>𝔽</sub>.
1. If _n_ is an integral Number, return _n_.
1. Return the greatest (closest to +∞) integral Number value that is not greater than _n_.
Expand Down Expand Up @@ -33877,7 +33870,7 @@ <h1>Math.sinh ( _x_ )</h1>
<p>It performs the following steps when called:</p>
<emu-alg>
1. Let _n_ be ? ToNumber(_x_).
1. If _n_ is not a finite Number or _n_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _n_.
1. If _n_ is not a non-zero finite Number, return _n_.
1. Return an implementation-approximated Number value representing the hyperbolic sine of ℝ(_n_).
</emu-alg>
<emu-note>
Expand Down Expand Up @@ -33977,7 +33970,7 @@ <h1>Math.trunc ( _x_ )</h1>
<p>It performs the following steps when called:</p>
<emu-alg>
1. Let _n_ be ? ToNumber(_x_).
1. If _n_ is not a finite Number or _n_ is either *+0*<sub>𝔽</sub> or *-0*<sub>𝔽</sub>, return _n_.
1. If _n_ is not a non-zero finite Number, return _n_.
1. If _n_ &lt; *1*<sub>𝔽</sub> and _n_ > *+0*<sub>𝔽</sub>, return *+0*<sub>𝔽</sub>.
1. If _n_ &lt; *-0*<sub>𝔽</sub> and _n_ > *-1*<sub>𝔽</sub>, return *-0*<sub>𝔽</sub>.
1. Return the integral Number nearest _n_ in the direction of *+0*<sub>𝔽</sub>.
Expand Down
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