1.1 Static Semantics: MV

A numeric literal stands for a value of the Number type or the BigInt type.

• NumericLiteralSeparator::_ has no MV.
• The MV of NumericLiteral::DecimalLiteral is the MV of DecimalLiteral.
• The MV of NonDecimalIntegerLiteral::BinaryIntegerLiteral is the MV of BinaryIntegerLiteral.
• The MV of NonDecimalIntegerLiteral::OctalIntegerLiteral is the MV of OctalIntegerLiteral.
• The MV of NonDecimalIntegerLiteral::HexIntegerLiteral is the MV of HexIntegerLiteral.
• The MV of DecimalLiteral::DecimalIntegerLiteral. is the MV of DecimalIntegerLiteral.
• The MV of DecimalLiteral::DecimalIntegerLiteral.DecimalDigits is the MV of DecimalIntegerLiteral plus (the MV of DecimalDigits × 10_ℝ^-_ℝn), where n is the mathematical value of the number of code points in DecimalDigits, excluding all occurrences of NumericLiteralSeparator.
• The MV of DecimalLiteral::DecimalIntegerLiteral.ExponentPart is the MV of DecimalIntegerLiteral × 10_ℝ^e, where e is the MV of ExponentPart.
• The MV of DecimalLiteral::DecimalIntegerLiteral.DecimalDigitsExponentPart is (the MV of DecimalIntegerLiteral plus (the MV of DecimalDigits × 10_ℝ^-_ℝn)) × 10_ℝ^e, where n is the mathematical integer number of code points in DecimalDigits, excluding all occurrences of NumericLiteralSeparator, and e is the MV of ExponentPart.
• The MV of DecimalLiteral::.DecimalDigits is the MV of DecimalDigits × 10_ℝ^-_ℝn, where n is the mathematical integer number of code points in DecimalDigits, excluding all occurrences of NumericLiteralSeparator.
• The MV of DecimalLiteral::.DecimalDigitsExponentPart is the MV of DecimalDigits × 10_ℝ^{e -_ℝ n}, where n is the mathematical integer number of code points in DecimalDigits, excluding all occurrences of NumericLiteralSeparator, and e is the MV of ExponentPart.
• The MV of DecimalLiteral::DecimalIntegerLiteral is the MV of DecimalIntegerLiteral.
• The MV of DecimalLiteral::DecimalIntegerLiteralExponentPart is the MV of DecimalIntegerLiteral × 10_ℝ^e, where e is the MV of ExponentPart.
• The MV of DecimalIntegerLiteral::0 is 0_ℝ.
• The MV of DecimalIntegerLiteral::NonZeroDigit is the MV of NonZeroDigit.
• The MV of DecimalIntegerLiteral::NonZeroDigitDecimalDigits is (the MV of NonZeroDigit × 10_ℝⁿ) plus the MV of DecimalDigits, where n is the mathematical integer number of code points in DecimalDigits, excluding all occurrences of NumericLiteralSeparator.
• The MV of DecimalDigits::DecimalDigit is the MV of DecimalDigit.
• The MV of DecimalDigits::DecimalDigitsDecimalDigit is (the MV of DecimalDigits × 10_ℝ) plus the MV of DecimalDigit.
• The MV of DecimalDigits::DecimalDigitsNumericLiteralSeparatorDecimalDigit is (the MV of DecimalDigits × 10_ℝ) plus the MV of DecimalDigit.
• The MV of ExponentPart::ExponentIndicatorSignedInteger is the MV of SignedInteger.
• The MV of SignedInteger::DecimalDigits is the MV of DecimalDigits.
• The MV of SignedInteger::+DecimalDigits is the MV of DecimalDigits.
• The MV of SignedInteger::-DecimalDigits is the negative of the MV of DecimalDigits.
• The MV of DecimalDigit::0 or of HexDigit::0 or of OctalDigit::0 or of BinaryDigit::0 is 0_ℝ.
• The MV of DecimalDigit::1 or of NonZeroDigit::1 or of HexDigit::1 or of OctalDigit::1 or of BinaryDigit::1 is 1_ℝ.
• The MV of DecimalDigit::2 or of NonZeroDigit::2 or of HexDigit::2 or of OctalDigit::2 is 2_ℝ.
• The MV of DecimalDigit::3 or of NonZeroDigit::3 or of HexDigit::3 or of OctalDigit::3 is 3_ℝ.
• The MV of DecimalDigit::4 or of NonZeroDigit::4 or of HexDigit::4 or of OctalDigit::4 is 4_ℝ.
• The MV of DecimalDigit::5 or of NonZeroDigit::5 or of HexDigit::5 or of OctalDigit::5 is 5_ℝ.
• The MV of DecimalDigit::6 or of NonZeroDigit::6 or of HexDigit::6 or of OctalDigit::6 is 6_ℝ.
• The MV of DecimalDigit::7 or of NonZeroDigit::7 or of HexDigit::7 or of OctalDigit::7 is 7_ℝ.
• The MV of DecimalDigit::8 or of NonZeroDigit::8 or of HexDigit::8 is 8_ℝ.
• The MV of DecimalDigit::9 or of NonZeroDigit::9 or of HexDigit::9 is 9_ℝ.
• The MV of HexDigit::a or of HexDigit::A is 10_ℝ.
• The MV of HexDigit::b or of HexDigit::B is 11_ℝ.
• The MV of HexDigit::c or of HexDigit::C is 12_ℝ.
• The MV of HexDigit::d or of HexDigit::D is 13_ℝ.
• The MV of HexDigit::e or of HexDigit::E is 14_ℝ.
• The MV of HexDigit::f or of HexDigit::F is 15_ℝ.
• The MV of BinaryIntegerLiteral::0bBinaryDigits is the MV of BinaryDigits.
• The MV of BinaryIntegerLiteral::0BBinaryDigits is the MV of BinaryDigits.
• The MV of BinaryDigits::BinaryDigit is the MV of BinaryDigit.
• The MV of BinaryDigits::BinaryDigitsNumericLiteralSeparatorBinaryDigit is (the MV of BinaryDigits × 2_ℝ) plus the MV of BinaryDigit.
• The MV of OctalIntegerLiteral::0oOctalDigits is the MV of OctalDigits.
• The MV of OctalIntegerLiteral::0OOctalDigits is the MV of OctalDigits.
• The MV of OctalDigits::OctalDigit is the MV of OctalDigit.
• The MV of OctalDigits::OctalDigitsOctalDigit is (the MV of OctalDigits × 8_ℝ) plus the MV of OctalDigit.
• The MV of OctalDigits::OctalDigitsNumericLiteralSeparatorOctalDigit is (the MV of OctalDigits × 8_ℝ) plus the MV of OctalDigit.
• The MV of HexIntegerLiteral::0xHexDigits is the MV of HexDigits.
• The MV of HexIntegerLiteral::0XHexDigits is the MV of HexDigits.
• The MV of HexDigits::HexDigit is the MV of HexDigit.
• The MV of HexDigits::HexDigitsHexDigit is (the MV of HexDigits × 16_ℝ) plus the MV of HexDigit.
• The MV of HexDigits::HexDigitsNumericLiteralSeparatorHexDigit is (the MV of HexDigits × 16_ℝ) plus the MV of HexDigit.
• The MV of Hex4Digits::HexDigitHexDigitHexDigitHexDigit is (0x1000_ℝ times the MV of the first HexDigit) plus (0x100_ℝ times the MV of the second HexDigit) plus (0x10_ℝ times the MV of the third HexDigit) plus the MV of the fourth HexDigit.

1.2 Static Semantics: NumericValue

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2.1 Runtime Semantics: MV

The conversion of a String to a Number value is similar overall to the determination of the Number value for a numeric literal (see 1), but some of the details are different, so the process for converting a String numeric literal to a value of Number type is given here. This value is determined in two steps: first, a mathematical value (MV) is derived from the String numeric literal; second, this mathematical value is rounded as described below. The MV on any grammar symbol, not provided below, is the MV for that symbol defined in 1.1.

• The MV of StringNumericLiteral:::[empty] is 0_ℝ.
• The MV of StringNumericLiteral:::StrWhiteSpace is 0_ℝ.
• The MV of StringNumericLiteral:::StrWhiteSpaceoptStrNumericLiteralStrWhiteSpaceopt is the MV of StrNumericLiteral, no matter whether white space is present or not.
• The MV of StrNumericLiteral::StrBinaryLiteral is the MV of StrBinaryLiteral.
• The MV of StrNumericLiteral::StrOctalLiteral is the MV of StrOctalLiteral.
• The MV of StrNumericLiteral::StrHexLiteral is the MV of StrHexLiteral.
• The MV of StrDecimalLiteral:::-StrUnsignedDecimalLiteral is the negative of the MV of StrUnsignedDecimalLiteral. (Note that if the MV of StrUnsignedDecimalLiteral is 0, the negative of this MV is also 0. The rounding rule described below handles the conversion of this signless mathematical zero to a floating-point +0 or -0 as appropriate.)
• The MV of StrUnsignedDecimalLiteral:::Infinity is 10_ℝ^10000_ℝ (a value so large that it will round to +∞).
• The MV of StrUnsignedDecimalLiteral:::DecimalDigits.DecimalDigits is the MV of the first DecimalDigits plus (the MV of the second DecimalDigits times 10_ℝ^-_ℝn), where n is the mathematical value of the number of code points in the second DecimalDigits.
• The MV of StrUnsignedDecimalLiteral:::DecimalDigits.ExponentPart is the MV of DecimalDigits times 10_ℝ^e, where e is the MV of ExponentPart.
• The MV of StrUnsignedDecimalLiteral:::DecimalDigits.DecimalDigitsExponentPart is (the MV of the first DecimalDigits plus (the MV of the second DecimalDigits times 10_ℝ^-_ℝn)) times 10_ℝ^e, 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 StrUnsignedDecimalLiteral:::.DecimalDigits is the MV of DecimalDigits times 10_ℝ^-_ℝn, where n is the mathematical value of the number of code points in DecimalDigits.
• The MV of StrUnsignedDecimalLiteral:::.DecimalDigitsExponentPart is the MV of DecimalDigits times 10_ℝ^{e -_ℝ n}, where n is the mathematical value of the number of code points in DecimalDigits and e is the MV of ExponentPart.
• The MV of StrUnsignedDecimalLiteral:::DecimalDigitsExponentPart is the MV of DecimalDigits times 10_ℝ^e, where e is the MV of ExponentPart.
• The MV of StrUnsignedDecimalLiteral:::StrDecimalDigits.StrDecimalDigits is the MV of the first StrDecimalDigits plus (the MV of the second StrDecimalDigits times 10_ℝ^-_ℝn), where n is the mathematical value of the number of code points in the second StrDecimalDigits.
• The MV of StrUnsignedDecimalLiteral:::StrDecimalDigits.StrExponentPart is the MV of StrDecimalDigits times 10_ℝ^e, where e is the MV of StrExponentPart.
• The MV of StrUnsignedDecimalLiteral:::StrDecimalDigits.StrDecimalDigitsStrExponentPart is (the MV of the first StrDecimalDigits plus (the MV of the second StrDecimalDigits times 10_ℝ^-_ℝn)) times 10_ℝ^e, where n is the mathematical value of the number of code points in the second StrDecimalDigits and e is the MV of StrExponentPart.
• The MV of StrUnsignedDecimalLiteral:::.StrDecimalDigits is the MV of StrDecimalDigits times 10_ℝ^-_ℝn, where n is the mathematical value of the number of code points in StrDecimalDigits.
• The MV of StrUnsignedDecimalLiteral:::.StrDecimalDigitsStrExponentPart is the MV of StrDecimalDigits times 10_ℝ^{e -_ℝ n}, where n is the mathematical value of the number of code points in StrDecimalDigits and e is the MV of StrExponentPart.
• The MV of StrUnsignedDecimalLiteral:::StrDecimalDigitsStrExponentPart is the MV of StrDecimalDigits times 10_ℝ^e, where e is the MV of StrExponentPart.
• The MV of StrExponentPart:::ExponentIndicatorStrSignedInteger is the MV of StrSignedInteger.
• The MV of StrSignedInteger:::StrDecimalDigits is the MV of StrDecimalDigits.
• The MV of StrSignedInteger:::+StrDecimalDigits is the MV of StrDecimalDigits.
• The MV of StrSignedInteger:::-StrDecimalDigits is the negative of the MV of StrDecimalDigits.
• The MV of StrBinaryLiteral:::0bStrBinaryDigits is the MV of StrBinaryDigits.
• The MV of StrBinaryLiteral:::0BStrBinaryDigits is the MV of StrBinaryDigits.
• The MV of StrBinaryDigits:::BinaryDigit is the MV of BinaryDigit.
• The MV of StrBinaryDigits:::StrBinaryDigitsBinaryDigit is (the MV of StrBinaryDigits × 2) plus the MV of BinaryDigit.
• The MV of StrOctalLiteral:::0oStrOctalDigits is the MV of StrOctalDigits.
• The MV of StrOctalLiteral:::0OStrOctalDigits is the MV of StrOctalDigits.
• The MV of StrOctalDigits:::OctalDigit is the MV of OctalDigit.
• The MV of StrOctalDigits:::StrOctalDigitsOctalDigit is (the MV of StrOctalDigits × 8) plus the MV of OctalDigit.
• The MV of StrHexLiteral:::0xStrHexDigits is the MV of StrHexDigits.
• The MV of StrHexLiteral:::0XStrHexDigits is the MV of StrHexDigits.
• The MV of StrHexDigits:::HexDigit is the MV of HexDigit.
• The MV of StrHexDigits:::StrHexDigitsHexDigit is (the MV of StrHexDigits × 16) plus the MV of HexDigit.

Once the exact MV for a String numeric literal has been determined, it is then rounded to a value of the Number type. If the MV is 0, then the rounded value is +0 unless the first non white space code point in the String numeric literal is -, in which case the rounded value is -0. Otherwise, the rounded value must be the Number value for the MV (in the sense defined in 6.1.6.1), unless the literal includes a StrUnsignedDecimalLiteral and the literal has more than 20 significant digits, in which case the Number value may be either the Number value for the MV of a literal produced by replacing each significant digit after the 20th with a 0 digit or the Number value for the MV of a literal produced by replacing each significant digit after the 20th with a 0 digit and then incrementing the literal at the 20th digit position. A digit is significant if it is not part of an ExponentPart and

• it is not 0; or
• there is a nonzero digit to its left and there is a nonzero digit, not in the ExponentPart, to its right.