An expression can be written as a single fraction: AsSingleFraction - #1622
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Entity.AsSingleFraction(), with the string extension and Transformation.AsSingleFraction, writes an expression as one numerator over one denominator, nothing divided inside either: 1 + 2/(1+t^2) is (t^2 + 3)/(t^2 + 1). Any expression, functions included, and nothing cancelled or multiplied out, so the denominator's factors stay and x/x stays x/x; a rational number counts as the fraction it is written as. Simplify never does it, and for a rational function in lowest terms Transformation.RationalCanonicalization is the canonical form. The name is the textbook's, "as a single fraction", as asked on #1239. Closes #1239. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Dividing by a fraction moves its denominator into the numerator, where it no longer stops the answer having a value: 1/(1/x) gathers to x, which is 0 at x = 0 where the expression has no value, and (a/b)/(c/d) to a d/(b c), which is 0 at d = 0. That is the one thing the simplification contract forbids (O4), so the answer now says so: x provided not x = 0. SingleQuotient.Of gains an overload that records each denominator it carries across; the one-argument form passes nothing and is unchanged for its callers. A denominator the new one still contains is not repeated, so (1/x)/(1/x) stays x / x. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
As it is by hand: 1/x + 1/x^2 is (x + 1)/x^2, where the product of the denominators put it over x^3 with a factor x in both halves. The common denominator is taken from the denominators as they are written -- each factor to the highest power any of them has it, and the whole numbers by their least common multiple -- so nothing is factorised to find it, and x^2 - 1 and x + 1 share nothing. SingleQuotient.OverLeastCommonDenominator is the entry AsSingleFraction uses; SingleQuotient.Of keeps the product for its callers, which divide out afterwards, and is unchanged for them. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
The tidying pass writes x^2 - 1 as x ^ 2 + -1, and Simplify's last step, NumericNeat, writes it back. So 1/(x^2 - 1) + 1/(x + 1) is (x + x ^ 2) / ((x ^ 2 - 1) * (x + 1)), and every other written form here is as it was. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Entity.AsSingleFraction(),Transformation.AsSingleFractionand"…".AsSingleFraction()write an expression as one numerator over one denominator, with nothing divided inside either. That is what a textbook calls writing it as a single fraction. The name is the one Happypig375 chose on #1239.sin(x) + 1/cos(x)is(1 + cos(x) * sin(x)) / cos(x).1/x + 1/x^2is(x + 1) / x ^ 2, not overx^3, and2/3 + x/2is over6. Nothing is factorised to find it, sox^2 - 1andx + 1share no factor.x/xstaysx / x, and1/(t^2+1) + 1/(t+1)keeps its denominator as(t ^ 2 + 1) * (t + 1). A rational number counts as the fraction it is:2/3 + x/2is(4 + 3 * x) / 6. Each half is tidied the waySimplifytidies, without its search: like terms are collected and operands put in order, so1/(x^2 - 1) + 1/(x + 1)is(x + x ^ 2) / ((x ^ 2 - 1) * (x + 1)).1/(1/x)isx provided not x = 0, and(a/b)/(c/d)isa * d / (b * c) provided not d = 0. This is O4 of the simplification contract. Where the new denominator still contains it, nothing is added:(1/x)/(1/x)isx / x.CanonicalizeAsRationalFunctionremains the canonical form for a rational function (lowest terms, denominator multiplied out and monic), and the docs point there. On #1239 I proposed modelling the two as one pipeline: single fraction, then lowest terms, then normalise, rather than one operation with a cost setting. The canonical form has the same domain gap on division by a fraction, filed as #1618, and the fix proposed there is to gather through this first step.SingleQuotient.Of(expr)keeps the product of the denominators, so its callers see no change: the integrator, partial fractions, the inequality solver and the quantifiers.SingleQuotient.OverLeastCommonDenominator(expr, carried)is the new entry this uses.The API is additive, so there is no BREAKING entry (as for #934).
Suite, net10.0, at
9e23f639on master8cb650e2: 14106 passed, 0 failed.Gate: allocation is what the baseline says on all 19 gated benchmarks.
Closes #1239.
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