From 6f480d3706c9f01f2f9044f4f016853a1e8239e6 Mon Sep 17 00:00:00 2001 From: Marshall Lochbaum Date: Fri, 8 Jan 2021 19:09:35 -0500 Subject: =?UTF-8?q?There's=20no=20identity=20element=20for=20=E2=88=BE?= =?UTF-8?q?=CB=9D=20on=20an=20empty=20list?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/bqn.js | 6 +++--- docs/spec/inferred.html | 2 +- 2 files changed, 4 insertions(+), 4 deletions(-) (limited to 'docs') diff --git a/docs/bqn.js b/docs/bqn.js index 747049b1..6612587d 100644 --- a/docs/bqn.js +++ b/docs/bqn.js @@ -204,9 +204,9 @@ let provide = [ ]; let runtime = run( - 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- ,provide.concat([0,1,2,32,3,8,-1,-Infinity,Infinity,4,5,-2,str("Β΄: 𝕩 must be a list"),str("Mapping: Equal-rank argument shapes don\'t agree"),str("Mapping: Argument shape prefixes don\'t agree"),str("⍋𝕩: 𝕩 must have rank at least 1"),str("/: Replication argument must have rank 1"),str("/: Amounts to replicate must be natural numbers"),str("βŒ½π•©: 𝕩 must have rank at least 1"),str("π•¨βŒ½π•©: 𝕨 must consist of integers"),str("↑𝕩: 𝕩 must have rank at least 1"),str("↓𝕩: 𝕩 must have rank at least 1"),str("βŠπ•©: 𝕩 must have rank at least 1"),str("βŠπ•©: 𝕩 cannot have length 0"),str("π•¨βŠπ•©: 𝕩 must be an array"),str("π•¨βŠπ•©: Indices in 𝕨 must be integers"),str("π•¨βŠπ•©: Indices out of range"),str(">𝕩: Elements of 𝕩 must have matching shapes"),str("βˆΎπ•©: Elements of 𝕩 must all have the same rank"),str("βˆΎπ•©: 𝕩 element rank must be at least argument rank"),str("βˆΎπ•©: 𝕩 element shapes must be compatible"),str("βˆΎπ•©: 𝕩 element trailing shapes must match"),str("βˆΎπ•©: 𝕩 must be an array"),str("𝕨"),str("𝕩: 𝕨 must "),str("have rank at most 1"),str("consist of integers"),str("↑"),str("↓"),str("Β« or Β»: 𝕩 must have rank at least 1"),str("Β« or Β»: 𝕨 must not have higher rank than 𝕩"),str("Β« or Β»: Rank of 𝕨 must be at least rank of 𝕩 minus 1"),str("Β« or Β»: 𝕨 must share 𝕩\'s major cell shape"),str("𝕨↕𝕩: 𝕩 must be an array"),str("𝕨↕𝕩: 𝕨 must have rank at most 1"),str("𝕨↕𝕩: Length of 𝕨 must be at most rank of 𝕩"),str("𝕨↕𝕩: 𝕨 must consist of natural numbers"),str("𝕨↕𝕩: Window length 𝕨 must be at most axis length plus one"),str("First-axis primitive: 𝕩 must have rank at least 1"),str("Multi-axis primitive: 𝕨 must have rank at most 1"),str("Multi-axis primitive: Length of 𝕨 must be at most rank of 𝕩"),str("β‰₯: No monadic form"),str("≀: No monadic form"),str("Β΄: Identity not found"),str("𝕨β₯Šπ•©: 𝕨 must have rank at most 1"),str("𝕨β₯Šπ•©: 𝕨 must consist of natural numbers"),str("𝕨β₯Šπ•©: Can\'t compute axis length when rest of shape is empty"),str("𝕨β₯Šπ•©: 𝕨 must consist of natural numbers or ∘ ⌊ ⌽ ↑"),str("𝕨β₯Šπ•©: Shape must be exact when reshaping with ∘"),str("↕𝕩: 𝕩 must consist of natural numbers"),str("↕𝕩: 𝕩 must be a number or list"),str("βŽ‰ or βš‡: 𝔽 result must have rank at most 1"),str("βŽ‰ or βš‡: 𝔽 result must have 1 to 3 elements"),str("βŽ‰ or βš‡: 𝔽 result must consist of integers"),str("˝: 𝕩 must have rank at least 1"),str("π•¨βˆΎπ•©: Rank of 𝕨 and 𝕩 must differ by at most 1"),str("π•¨βˆΎπ•©: Cell shapes of 𝕨 and 𝕩 must match"),str("𝕨/𝕩: Lengths of components of 𝕨 must match 𝕩"),str("βŠ”: Grouping argument must consist of integers"),str("βŠ”: Grouping argument values cannot be less than Β―1"),str("βŠ”π•©: 𝕩 must be a list"),str("π•¨βŠ”π•©: 𝕩 must be an array"),str("π•¨βŠ”π•©: Compound 𝕨 must be a list"),str("π•¨βŠ”π•©: Total rank of 𝕨 must be at most rank of 𝕩"),str("π•¨βŠ”π•©: Lengths of components of 𝕨 must be compatible with 𝕩"),str("π•¨βŠ‘π•©: Indices in compound 𝕨 must be lists"),str("π•¨βŠ‘π•©: Index length in 𝕨 must match rank of 𝕩"),str("π•¨βŠ‘π•©: Indices in 𝕨 must consist of integers"),str("π•¨βŠ‘π•©: Index out of range"),str("⍋ or ⍒: Rank of 𝕨 must be at least 1"),str("⍋ or ⍒: Rank of 𝕩 must be at least cell rank of 𝕨"),str("⍋ or ⍒: 𝕨 must be sorted"),str("pβŠπ•© or π•¨βˆŠp: p must have rank at least 1"),str("βˆŠπ•© or βŠπ•©: 𝕩 must have rank at least 1"),str("⍷𝕩: Rank of 𝕨 cannot exceed rank of 𝕩"),str("𝕨⍉𝕩: 𝕨 must have rank at most 1"),str("𝕨⍉𝕩: Length of 𝕨 must not exceed rank of 𝕩"),str("𝕨⍉𝕩: 𝕨 must consist of natural numbers"),str("𝕨⍉𝕩: Skipped result axis"),str("⍟: Repetition numbers in 𝕨 must be integers"),str("⁼: Inverse failed"),str("⁼: Inverse does not exist"),str("⁼: Inverse not found"),str("Cannot currently invert blocks"),str("Cannot invert modifier")]) - 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0,2,7,21,3,7,0,19,8,8,21,1,7,17,21,1,5,21,0,2,0,10,21,1,7,17,7,22,1,6,13,14,0,24,22,1,7,12,14,21,0,1,21,3,23,16,25,21,0,2,0,10,22,1,7,13,14,21,0,2,25,0,24,22,1,9,12,25,21,1,3,0,9,21,1,4,21,3,23,16,17,0,19,16,0,20,0,8,21,3,7,21,1,3,8,21,3,4,0,18,7,21,1,4,19,7,16,22,0,3,11,21,3,27,22,1,10,13,14,21,0,3,21,3,25,0,10,7,21,1,7,17,0,14,21,3,7,0,24,8,0,9,0,24,19,21,3,9,15,206,8,16,14,21,0,1,0,17,16,21,3,8,0,18,8,22,0,4,11,14,21,1,9,21,3,1,21,3,9,15,207,8,16,14,21,1,6,0,17,21,1,10,17,0,20,21,0,4,7,16,25,21,0,1,0,19,16,0,20,0,9,21,3,7,21,0,1,8,21,3,4,0,18,7,21,1,3,19,7,16,25,21,0,1,0,19,16,0,20,0,10,21,3,7,21,0,2,8,7,16,25,21,0,1,0,24,0,9,21,1,3,17,21,3,18,21,0,2,17,21,3,9,21,3,65,21,0,0,21,3,7,21,0,2,0,8,0,24,17,8,7,8,16,21,2,3,21,0,2,21,1,4,16,17,25,21,2,7,21,3,83,21,0,1,17,0,19,16,0,20,21,2,6,0,3,16,7,16,21,3,27,22,2,6,13,14,21,2,7,25,21,0,1,21,3,70,21,2,7,17,0,20,21,2,6,21,3,8,0,18,8,7,16,25,21,0,2,21,3,78,16,21,3,76,0,23,17,0,7,0,88,17,14,21,0,2,21,3,78,16,0,19,16,21,3,76,0,23,17,21,3,75,21,0,2,17,25,21,0,2,0,20,21,0,1,21,3,8,0,18,8,7,16,25,21,0,1,21,3,79,21,0,2,17,0,21,21,3,73,7,16,21,3,25,21,3,82,7,16,22,0,3,11,14,21,0,3,21,3,100,16,0,20,21,0,2,21,3,8,0,18,8,7,16,21,3,25,21,3,75,7,16,22,0,4,11,14,21,0,3,21,3,77,21,3,7,21,1,4,8,21,3,9,21,0,2,21,0,1,3,2,21,3,8,21,3,48,0,18,7,8,21,3,5,15,208,8,8,16,14,21,0,4,25,21,1,4,21,3,82,21,3,7,21,0,1,8,21,3,6,21,2,5,8,21,1,3,17,21,3,79,21,3,7,0,23,8,21,3,9,21,0,1,21,3,82,0,24,17,21,3,5,21,1,5,8,8,16,25,21,0,1,0,24,21,3,8,21,3,77,8,21,3,9,15,209,8,21,0,2,17,25,21,0,2,21,3,118,16,21,3,101,16,22,0,3,11,14,21,0,1,21,3,117,21,0,2,21,3,29,21,0,3,17,17,21,3,4,21,3,83,7,0,24,17,21,3,72,0,23,17,21,3,4,21,3,29,7,21,0,3,17,22,0,4,11,14,21,0,1,21,1,3,21,3,116,21,3,51,8,21,0,2,21,3,29,21,0,4,17,17,21,2,2,21,3,9,21,3,75,21,3,7,21,3,83,21,3,7,21,0,4,8,8,21,3,82,21,3,2,19,8,21,0,2,21,3,78,16,17,25,21,3,2,21,0,1,21,0,4,19,25,21,0,1,21,1,8,0,9,0,24,17,21,4,9,21,1,9,0,9,0,24,19,21,4,9,15,210,8,8,0,24,0,9,21,0,1,17,17,25,21,0,2,0,8,21,1,3,17,22,1,4,12,14,21,0,1,0,8,21,1,3,17,22,1,7,12,21,2,6,16,22,1,6,12,0,8,21,1,3,17,21,2,6,16,22,1,5,12,14,21,1,7,0,15,21,3,6,17,22,1,8,12,14,21,1,6,21,1,11,16,14,21,1,7,21,1,13,21,1,4,17,25,21,0,2,0,8,0,24,17,22,1,4,12,0,14,21,1,6,17,0,10,0,25,17,0,8,22,1,8,13,14,21,0,2,25,21,0,1,0,8,0,24,17,22,1,7,12,0,14,21,1,5,17,0,8,22,1,8,13,14,21,0,1,25,21,0,1,21,1,10,21,4,0,21,1,8,8,21,0,2,17,25,21,0,1,0,18,21,1,7,21,1,13,21,1,4,17,17,25,0,24,0,8,21,0,1,17,21,2,4,16,22,2,6,12,21,2,5,16,22,2,7,12,14,21,2,6,25,21,0,1,0,19,16,21,2,5,21,0,1,7,22,2,6,13,25,21,4,3,21,0,1,0,3,16,7,21,1,4,3,2,21,4,0,0,15,21,4,7,0,23,8,8,22,1,4,12,25,21,0,1,21,4,25,21,4,83,7,16,21,4,75,16,22,2,3,12,14,21,0,1,21,4,25,21,4,71,7,16,21,4,75,21,1,4,17,22,1,4,12,25,0,25,21,4,67,21,0,2,17,21,4,71,16,22,0,3,11,21,4,82,21,0,1,17,22,0,4,11,21,3,2,16,22,0,5,11,14,21,0,1,21,0,4,3,2,0,18,21,0,5,17,21,2,3,21,0,2,21,4,70,0,25,17,21,4,75,21,0,5,17,21,4,82,21,0,3,17,17,25,21,0,1,14,0,25,22,2,8,12,14,21,2,4,22,2,7,12,14,21,0,1,22,2,4,12,25] + ,provide.concat([0,1,2,32,3,8,-1,-Infinity,Infinity,4,5,-2,str("Β΄: 𝕩 must be a list"),str("Mapping: Equal-rank argument shapes don\'t agree"),str("Mapping: Argument shape prefixes don\'t agree"),str("⍋𝕩: 𝕩 must have rank at least 1"),str("/: Replication argument must have rank 1"),str("/: Amounts to replicate must be natural numbers"),str("βŒ½π•©: 𝕩 must have rank at least 1"),str("π•¨βŒ½π•©: 𝕨 must consist of integers"),str("↑𝕩: 𝕩 must have rank at least 1"),str("↓𝕩: 𝕩 must have rank at least 1"),str("βŠπ•©: 𝕩 must have rank at least 1"),str("βŠπ•©: 𝕩 cannot have length 0"),str("π•¨βŠπ•©: 𝕩 must be an array"),str("π•¨βŠπ•©: Indices in 𝕨 must be integers"),str("π•¨βŠπ•©: Indices out of range"),str(">𝕩: Elements of 𝕩 must have matching shapes"),str("βˆΎπ•©: Elements of 𝕩 must all have the same rank"),str("βˆΎπ•©: 𝕩 element rank must be at least argument rank"),str("βˆΎπ•©: 𝕩 element shapes must be compatible"),str("βˆΎπ•©: 𝕩 element trailing shapes must match"),str("βˆΎπ•©: 𝕩 must be an array"),str("𝕨"),str("𝕩: 𝕨 must "),str("have rank at most 1"),str("consist of integers"),str("↑"),str("↓"),str("Β« or Β»: 𝕩 must have rank at least 1"),str("Β« or Β»: 𝕨 must not have higher rank than 𝕩"),str("Β« or Β»: Rank of 𝕨 must be at least rank of 𝕩 minus 1"),str("Β« or Β»: 𝕨 must share 𝕩\'s major cell shape"),str("𝕨↕𝕩: 𝕩 must be an array"),str("𝕨↕𝕩: 𝕨 must have rank at most 1"),str("𝕨↕𝕩: Length of 𝕨 must be at most rank of 𝕩"),str("𝕨↕𝕩: 𝕨 must consist of natural numbers"),str("𝕨↕𝕩: Window length 𝕨 must be at most axis length plus one"),str("First-axis primitive: 𝕩 must have rank at least 1"),str("Multi-axis primitive: 𝕨 must have rank at most 1"),str("Multi-axis primitive: Length of 𝕨 must be at most rank of 𝕩"),str("β‰₯: No monadic form"),str("≀: No monadic form"),str("Β΄: Identity not found"),str("𝕨β₯Šπ•©: 𝕨 must have rank at most 1"),str("𝕨β₯Šπ•©: 𝕨 must consist of natural numbers"),str("𝕨β₯Šπ•©: Can\'t compute axis length when rest of shape is empty"),str("𝕨β₯Šπ•©: 𝕨 must consist of natural numbers or ∘ ⌊ ⌽ ↑"),str("𝕨β₯Šπ•©: Shape must be exact when reshaping with ∘"),str("↕𝕩: 𝕩 must consist of natural numbers"),str("↕𝕩: 𝕩 must be a number or list"),str("βŽ‰ or βš‡: 𝔽 result must have rank at most 1"),str("βŽ‰ or βš‡: 𝔽 result must have 1 to 3 elements"),str("βŽ‰ or βš‡: 𝔽 result must consist of integers"),str("˝: 𝕩 must have rank at least 1"),str("˝: Identity does not exist"),str("π•¨βˆΎπ•©: Rank of 𝕨 and 𝕩 must differ by at most 1"),str("π•¨βˆΎπ•©: Cell shapes of 𝕨 and 𝕩 must match"),str("𝕨/𝕩: Lengths of components of 𝕨 must match 𝕩"),str("βŠ”: Grouping argument must consist of integers"),str("βŠ”: Grouping argument values cannot be less than Β―1"),str("βŠ”π•©: 𝕩 must be a list"),str("π•¨βŠ”π•©: 𝕩 must be an array"),str("π•¨βŠ”π•©: Compound 𝕨 must be a list"),str("π•¨βŠ”π•©: Total rank of 𝕨 must be at most rank of 𝕩"),str("π•¨βŠ”π•©: Lengths of components of 𝕨 must be compatible with 𝕩"),str("π•¨βŠ‘π•©: Indices in compound 𝕨 must be lists"),str("π•¨βŠ‘π•©: Index length in 𝕨 must match rank of 𝕩"),str("π•¨βŠ‘π•©: Indices in 𝕨 must consist of integers"),str("π•¨βŠ‘π•©: Index out of range"),str("⍋ or ⍒: Rank of 𝕨 must be at least 1"),str("⍋ or ⍒: Rank of 𝕩 must be at least cell rank of 𝕨"),str("⍋ or ⍒: 𝕨 must be sorted"),str("pβŠπ•© or π•¨βˆŠp: p must have rank at least 1"),str("βˆŠπ•© or βŠπ•©: 𝕩 must have rank at least 1"),str("⍷𝕩: Rank of 𝕨 cannot exceed rank of 𝕩"),str("𝕨⍉𝕩: 𝕨 must have rank at most 1"),str("𝕨⍉𝕩: Length of 𝕨 must not exceed rank of 𝕩"),str("𝕨⍉𝕩: 𝕨 must consist of natural numbers"),str("𝕨⍉𝕩: Skipped result axis"),str("⍟: Repetition numbers in 𝕨 must be integers"),str("⁼: Inverse failed"),str("⁼: Inverse does not exist"),str("⁼: Inverse not found"),str("Cannot currently invert blocks"),str("Cannot invert modifier")]) + ,[[0,1,0,166],[2,0,3740,6],[0,0,3767,3],[0,0,3771,3],[0,0,3775,3],[1,0,3779,5],[1,0,3787,5],[2,0,3805,6],[2,0,3820,6],[2,0,3839,6],[2,0,3861,6],[2,0,3883,6],[0,0,3915,3],[0,0,3921,3],[0,0,3930,3],[0,0,3936,3],[0,0,3947,3],[0,0,3956,3],[1,0,3965,9],[0,0,4078,3],[0,0,4118,4],[0,0,4194,5],[0,0,4269,3],[1,1,4286,4],[1,1,4359,3],[0,0,4425,3],[0,0,4436,5],[2,0,4520,12],[0,0,4726,4],[0,0,4775,6],[0,0,4865,7],[0,0,4994,4],[0,0,5055,4],[0,0,5157,3],[0,0,5210,4],[1,1,5271,2],[0,0,5304,3],[0,0,5354,4],[2,0,5496,16],[1,0,5727,5],[0,0,5758,3],[0,0,5785,3],[0,0,5811,3],[0,0,5850,4],[0,0,5947,7],[0,0,5994,14],[0,0,6228,3],[1,1,6258,8],[1,0,6311,6],[1,1,6419,2],[0,0,6477,5],[0,0,6639,5],[0,0,6776,7],[0,0,6945,3],[0,0,7004,4],[1,0,7078,5],[2,1,7095,4],[0,0,7131,3],[0,0,7158,3],[0,0,7180,3],[0,0,7202,3],[0,0,7239,9],[0,0,7390,5],[0,0,7424,3],[1,1,7519,2],[2,0,7588,10],[2,0,7647,8],[1,0,7840,7],[0,0,7907,8],[0,0,8143,5],[0,0,8209,3],[1,0,8243,8],[0,0,8342,4],[0,0,8425,6],[0,0,8749,4],[0,0,8962,3],[0,0,8980,4],[1,0,9052,9],[1,1,9121,3],[1,0,9145,14],[1,1,9583,4],[1,0,9632,6],[0,0,9771,4],[0,0,9839,3],[0,0,9913,3],[0,0,9968,4],[0,0,10097,4],[0,0,10217,3],[2,0,10274,6],[1,1,10302,2],[2,0,10310,14],[2,0,10497,7],[0,0,10538,6],[0,0,10688,3],[2,1,10758,4],[0,0,10841,3],[0,0,10879,5],[0,0,11023,3],[0,0,11070,3],[0,0,11096,3],[1,0,11183,5],[2,0,11204,6],[0,0,11219,3],[0,0,11227,3],[1,1,11248,2],[0,0,11277,3],[0,0,11313,3],[0,0,11361,3],[0,0,11397,3],[1,1,11484,2],[0,0,11517,3],[0,0,11528,4],[0,0,11565,3],[0,0,11580,3],[0,0,11609,3],[0,0,11624,3],[0,0,11642,3],[0,0,11657,3],[0,0,11667,3],[0,0,11698,7],[0,0,11777,3],[1,1,11802,2],[1,1,11884,2],[1,1,11888,2],[0,0,11892,3],[0,0,11904,3],[0,0,11927,3],[1,0,11948,6],[1,0,12065,11],[1,0,12286,5],[0,0,12301,3],[0,0,12315,3],[0,0,12329,3],[1,0,12374,8],[0,0,12512,8],[0,0,12679,3],[0,0,12705,3],[0,0,12711,4],[0,0,12760,3],[0,0,12770,3],[1,0,12814,7],[0,0,12840,9],[1,0,13126,5],[0,0,13142,7],[0,0,13309,11],[0,0,13545,8],[0,0,13857,3],[0,0,13885,5],[0,0,13980,8],[0,0,14235,3],[0,0,14275,3],[0,0,14295,3],[1,0,14349,6],[0,0,14502,6],[0,0,14624,5],[0,0,14701,3],[0,0,14724,3],[0,0,14735,3],[0,0,14756,3],[0,0,14787,3],[0,0,14825,6],[0,0,14930,6],[0,0,14979,4],[0,0,15008,3],[0,0,15027,4],[1,1,15184,2],[0,0,15188,3],[0,0,15231,3],[0,0,15242,3],[0,0,15250,3],[0,0,15272,3],[0,0,15280,3],[0,0,15292,3],[0,0,15315,3],[0,0,15342,3],[1,0,15360,5],[1,0,15371,5],[0,0,15382,14],[0,0,15552,3],[0,0,15572,3],[0,0,15601,4],[0,0,15624,3],[1,0,15657,5],[0,0,15683,3],[0,0,15753,3],[0,1,15767,0],[0,0,15774,5],[0,0,15907,3],[0,0,15937,3],[0,0,15957,3],[0,0,16011,3],[0,0,16046,3],[0,0,16070,3],[0,0,16113,3],[0,0,16130,5],[0,0,16240,3],[0,0,16295,3],[0,0,16318,5],[1,0,16447,5],[0,0,16458,3],[0,0,16497,3],[0,0,16580,3],[0,0,16614,3],[0,0,16643,3],[0,0,16661,3],[0,0,16678,3],[0,0,16707,3],[0,0,16725,3],[0,0,16757,3],[0,0,16800,6],[0,0,16879,3]] ); // Compiler diff --git a/docs/spec/inferred.html b/docs/spec/inferred.html index d4d22610..e1d08a02 100644 --- a/docs/spec/inferred.html +++ b/docs/spec/inferred.html @@ -66,7 +66,7 @@ -

Additionally, the identity of ∾˝ must be recognized: if 0=≠𝕩, then βˆΎΛπ•© is (0∾2↓≒𝕩)β₯Šπ•©.

+

Additionally, the identity of ∾˝ must be recognized: if 0=≠𝕩 and 1<=𝕩, then βˆΎΛπ•© is (0∾2↓≒𝕩)β₯Šπ•©. If 1==𝕩, then there is no identity element, as the result of ∾ always has rank at least 1, but the cell rank is 0.

Undo

The Undo 1-modifier ⁼, given an operand 𝔽 and argument 𝕩, and possibly a left argument 𝕨, finds a value y such that 𝕩≑𝕨𝔽y, that is, an element of the pre-image of 𝕩 under 𝔽 or π•¨π”½βŠ’. Thus it satisfies the constraint 𝕩 ≑ 𝕨𝔽𝕨𝔽⁼𝕩 (π•¨π”½βΌβŠ’ is a right inverse of π•¨π”½βŠ’) provided 𝔽⁼ and 𝔽 both complete without error. 𝔽⁼ should of course give an error if no inverse element exists, and can also fail if no inverse can be found. It is also preferred for 𝔽⁼ to give an error if there are many choices of inverse with no clear way to choose one of them: for example, 0β€Ώ0⍉m returns the diagonal of matrix m; 0β€Ώ0⍉⁼2β€Ώ3 requires values to be chosen for the off-diagonal elements in its result. It is better to give an error, encouraging the programmer to use a fully-specified approach like 2β€Ώ3⌾(0β€Ώ0βŠΈβ‰) applied to a matrix of initial elements, than to return a result that could be very different from other implementations.

When working with limited-precision numbers, it may be difficult or impossible to exactly invert the operand function. Instead, it is generally acceptable to perform a computation that, if done with unlimited precision, would exactly invert 𝔽 computed with unlimited precision. This principle is the basis for the numeric inverses specified below. It is also acceptable to find an inverse by numeric methods, provided that the error in the inverse value found relative to an unlimited-precision inverse can be kept close to the inherent error in the implementation's number format.

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