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Rafael-SOWNet merged 1 commit into from
Oct 31, 2026
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A Taylor expansion simplifies each derivative before taking the next #2302

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17 changes: 17 additions & 0 deletions BREAKING-CHANGES.md
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Original file line number Diff line number Diff line change
Expand Up @@ -870,6 +870,23 @@ reals, to `1` and `0` ([#2195](https://github.com/asc-community/AngouriMath/issu
| `"arg(x)".ToEntity().Compile("x").Call(-2)` | throws `UncompilableNodeException`, `arg` being a variable | `pi` |
| `"conj(x)".ToEntity().Compile<Complex, Complex>("x")(3 + 4i)` | throws `UncompilableNodeException`, `conj` being a variable | `3 - 4i` |

### A Taylor expansion simplifies each derivative before taking the next

**Silent: `MathS.Series.Taylor`, `Maclaurin` and `TaylorTerms` print simplified coefficients, an expansion where the
function has no value is `NaN`, and no terms are `0`.** Each derivative was taken from the one before it as it stood,
and only the term built from it was simplified. A quotient's derivatives grew about eightfold a step: the ninth of
`sin(x) / x` had 7,797,110 nodes, and 87 simplified. One derivative more than the terms kept was also taken, then
dropped. An expansion around a point where the function has no value summed a `NaN` for each term
([#2291](https://github.com/asc-community/AngouriMath/issues/2291)).

| Input | Was (2.6.0) | Now |
|---|---|---|
| `MathS.Series.Taylor("sin(x) / x", 11, ("x", 1))` | `OutOfMemoryException` after 32 s with a 2 GB heap | 529 characters, 0.9 s |
| `MathS.Series.Taylor("sin(x) / x", 7, ("x", 1))` | 36,421 characters | `sin(1) + (cos(1) - sin(1)) * (x - 1) + ((-2) * cos(1) + sin(1)) * (x - 1) ^ 2 / 2! + ...` |
| `MathS.Series.Taylor("sin(x) / x", 4, ("x", 0))` | `NaN + NaN + NaN / 2! + NaN / 3!` | `NaN` |
| `MathS.Series.Taylor("sin(x)", 0, ("x", 1))` | throws `AngouriBugException` | `0` |
| `MathS.Series.Taylor("sin(x) + cos(y)", 6, ("x", 1))` | `sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + ...` | `cos(y) + sin(1) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + -cos(1) * (x - 1) ^ 3 / 3! + ...` |

### A logarithm to base 0 has no value

**Silent: `log(0, x)` is `NaN`, where it was `0` for a positive `x`.** Evaluation and both compilers took it as
Expand Down
64 changes: 32 additions & 32 deletions Sources/AngouriMath/Convenience/MathS.cs
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Expand Up @@ -2681,23 +2681,23 @@ public static class Series
/// x ^ 9 / 362880 - x ^ 7 / 5040 + x ^ 5 / 120 - x ^ 3 / 6 + x
/// ----------------------
/// sin(x) + cos(y)
/// 1 + (6 * x ^ 5 - 120 * x ^ 3) / 720 + x + (2 * y ^ 4 - 24 * y ^ 2) / 48
/// x ^ 5 / 120 - x ^ 3 / 6 + x + 1 + y ^ 4 / 24 - y ^ 2 / 2
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + -sin(1) * (-1) * (x - 1) ^ 4 / 4! + cos(1) * (-1) * (-1) * (x - 1) ^ 5 / 5!
/// cos(y) + sin(1) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + -cos(1) * (x - 1) ^ 3 / 3! + sin(1) * (x - 1) ^ 4 / 4! + cos(1) * (x - 1) ^ 5 / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + cos(5) * (-1) * (y - 5) ^ 2) / 2! + (cos(1) * (-1) * (x - 1) ^ 3 + -sin(5) * (-1) * (y - 5) ^ 3) / 3! + (-sin(1) * (-1) * (x - 1) ^ 4 + cos(5) * (-1) * (-1) * (y - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (x - 1) ^ 5 + -sin(5) * (-1) * (-1) * (y - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + -cos(5) * (y - 5) ^ 2) / 2! + (-cos(1) * (x - 1) ^ 3 + sin(5) * (y - 5) ^ 3) / 3! + (sin(1) * (x - 1) ^ 4 + cos(5) * (y - 5) ^ 4) / 4! + (cos(1) * (x - 1) ^ 5 + -sin(5) * (y - 5) ^ 5) / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + cos(5) * (-1) * (z_2 - 5) ^ 2) / 2! + (cos(1) * (-1) * (z_1 - 1) ^ 3 + -sin(5) * (-1) * (z_2 - 5) ^ 3) / 3! + (-sin(1) * (-1) * (z_1 - 1) ^ 4 + cos(5) * (-1) * (-1) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (z_1 - 1) ^ 5 + -sin(5) * (-1) * (-1) * (z_2 - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + -cos(5) * (z_2 - 5) ^ 2) / 2! + (-cos(1) * (z_1 - 1) ^ 3 + sin(5) * (z_2 - 5) ^ 3) / 3! + (sin(1) * (z_1 - 1) ^ 4 + cos(5) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (z_1 - 1) ^ 5 + -sin(5) * (z_2 - 5) ^ 5) / 5!
/// ----------------------
/// Received cos(1)
/// Received x + -sin(1) * (y - 1)
/// Received cos(1) * (-1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + -sin(1) * (-1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (-1) * (-1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (-1) * (-1) * (y - 1) ^ 5) / 5!
/// Received -cos(1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + sin(1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (y - 1) ^ 5) / 5!
/// </code>
/// </example>
public static Entity Maclaurin(Entity expr, int degree, params Variable[] exprVariables)
Expand Down Expand Up @@ -2769,23 +2769,23 @@ public static Entity Maclaurin(Entity expr, int degree, params Variable[] exprVa
/// x ^ 9 / 362880 - x ^ 7 / 5040 + x ^ 5 / 120 - x ^ 3 / 6 + x
/// ----------------------
/// sin(x) + cos(y)
/// 1 + (6 * x ^ 5 - 120 * x ^ 3) / 720 + x + (2 * y ^ 4 - 24 * y ^ 2) / 48
/// x ^ 5 / 120 - x ^ 3 / 6 + x + 1 + y ^ 4 / 24 - y ^ 2 / 2
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + -sin(1) * (-1) * (x - 1) ^ 4 / 4! + cos(1) * (-1) * (-1) * (x - 1) ^ 5 / 5!
/// cos(y) + sin(1) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + -cos(1) * (x - 1) ^ 3 / 3! + sin(1) * (x - 1) ^ 4 / 4! + cos(1) * (x - 1) ^ 5 / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + cos(5) * (-1) * (y - 5) ^ 2) / 2! + (cos(1) * (-1) * (x - 1) ^ 3 + -sin(5) * (-1) * (y - 5) ^ 3) / 3! + (-sin(1) * (-1) * (x - 1) ^ 4 + cos(5) * (-1) * (-1) * (y - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (x - 1) ^ 5 + -sin(5) * (-1) * (-1) * (y - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + -cos(5) * (y - 5) ^ 2) / 2! + (-cos(1) * (x - 1) ^ 3 + sin(5) * (y - 5) ^ 3) / 3! + (sin(1) * (x - 1) ^ 4 + cos(5) * (y - 5) ^ 4) / 4! + (cos(1) * (x - 1) ^ 5 + -sin(5) * (y - 5) ^ 5) / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + cos(5) * (-1) * (z_2 - 5) ^ 2) / 2! + (cos(1) * (-1) * (z_1 - 1) ^ 3 + -sin(5) * (-1) * (z_2 - 5) ^ 3) / 3! + (-sin(1) * (-1) * (z_1 - 1) ^ 4 + cos(5) * (-1) * (-1) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (z_1 - 1) ^ 5 + -sin(5) * (-1) * (-1) * (z_2 - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + -cos(5) * (z_2 - 5) ^ 2) / 2! + (-cos(1) * (z_1 - 1) ^ 3 + sin(5) * (z_2 - 5) ^ 3) / 3! + (sin(1) * (z_1 - 1) ^ 4 + cos(5) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (z_1 - 1) ^ 5 + -sin(5) * (z_2 - 5) ^ 5) / 5!
/// ----------------------
/// Received cos(1)
/// Received x + -sin(1) * (y - 1)
/// Received cos(1) * (-1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + -sin(1) * (-1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (-1) * (-1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (-1) * (-1) * (y - 1) ^ 5) / 5!
/// Received -cos(1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + sin(1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (y - 1) ^ 5) / 5!
/// </code>
/// </example>
public static Entity Taylor(Entity expr, int degree, params (Variable exprVariable, Entity point)[] exprVariables)
Expand Down Expand Up @@ -2858,23 +2858,23 @@ public static Entity Taylor(Entity expr, int degree, params (Variable exprVariab
/// x ^ 9 / 362880 - x ^ 7 / 5040 + x ^ 5 / 120 - x ^ 3 / 6 + x
/// ----------------------
/// sin(x) + cos(y)
/// 1 + (6 * x ^ 5 - 120 * x ^ 3) / 720 + x + (2 * y ^ 4 - 24 * y ^ 2) / 48
/// x ^ 5 / 120 - x ^ 3 / 6 + x + 1 + y ^ 4 / 24 - y ^ 2 / 2
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + -sin(1) * (-1) * (x - 1) ^ 4 / 4! + cos(1) * (-1) * (-1) * (x - 1) ^ 5 / 5!
/// cos(y) + sin(1) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + -cos(1) * (x - 1) ^ 3 / 3! + sin(1) * (x - 1) ^ 4 / 4! + cos(1) * (x - 1) ^ 5 / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + cos(5) * (-1) * (y - 5) ^ 2) / 2! + (cos(1) * (-1) * (x - 1) ^ 3 + -sin(5) * (-1) * (y - 5) ^ 3) / 3! + (-sin(1) * (-1) * (x - 1) ^ 4 + cos(5) * (-1) * (-1) * (y - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (x - 1) ^ 5 + -sin(5) * (-1) * (-1) * (y - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + -cos(5) * (y - 5) ^ 2) / 2! + (-cos(1) * (x - 1) ^ 3 + sin(5) * (y - 5) ^ 3) / 3! + (sin(1) * (x - 1) ^ 4 + cos(5) * (y - 5) ^ 4) / 4! + (cos(1) * (x - 1) ^ 5 + -sin(5) * (y - 5) ^ 5) / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + cos(5) * (-1) * (z_2 - 5) ^ 2) / 2! + (cos(1) * (-1) * (z_1 - 1) ^ 3 + -sin(5) * (-1) * (z_2 - 5) ^ 3) / 3! + (-sin(1) * (-1) * (z_1 - 1) ^ 4 + cos(5) * (-1) * (-1) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (z_1 - 1) ^ 5 + -sin(5) * (-1) * (-1) * (z_2 - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + -cos(5) * (z_2 - 5) ^ 2) / 2! + (-cos(1) * (z_1 - 1) ^ 3 + sin(5) * (z_2 - 5) ^ 3) / 3! + (sin(1) * (z_1 - 1) ^ 4 + cos(5) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (z_1 - 1) ^ 5 + -sin(5) * (z_2 - 5) ^ 5) / 5!
/// ----------------------
/// Received cos(1)
/// Received x + -sin(1) * (y - 1)
/// Received cos(1) * (-1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + -sin(1) * (-1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (-1) * (-1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (-1) * (-1) * (y - 1) ^ 5) / 5!
/// Received -cos(1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + sin(1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (y - 1) ^ 5) / 5!
/// </code>
/// </example>
public static Entity Taylor(Entity expr, int degree, params (Variable exprVariable, Variable polyVariable, Entity point)[] exprToPolyVars)
Expand Down Expand Up @@ -2948,23 +2948,23 @@ public static Entity Taylor(Entity expr, int degree, params (Variable exprVariab
/// x ^ 9 / 362880 - x ^ 7 / 5040 + x ^ 5 / 120 - x ^ 3 / 6 + x
/// ----------------------
/// sin(x) + cos(y)
/// 1 + (6 * x ^ 5 - 120 * x ^ 3) / 720 + x + (2 * y ^ 4 - 24 * y ^ 2) / 48
/// x ^ 5 / 120 - x ^ 3 / 6 + x + 1 + y ^ 4 / 24 - y ^ 2 / 2
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(y) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + cos(1) * (-1) * (x - 1) ^ 3 / 3! + -sin(1) * (-1) * (x - 1) ^ 4 / 4! + cos(1) * (-1) * (-1) * (x - 1) ^ 5 / 5!
/// cos(y) + sin(1) + cos(1) * (x - 1) + -sin(1) * (x - 1) ^ 2 / 2! + -cos(1) * (x - 1) ^ 3 / 3! + sin(1) * (x - 1) ^ 4 / 4! + cos(1) * (x - 1) ^ 5 / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + cos(5) * (-1) * (y - 5) ^ 2) / 2! + (cos(1) * (-1) * (x - 1) ^ 3 + -sin(5) * (-1) * (y - 5) ^ 3) / 3! + (-sin(1) * (-1) * (x - 1) ^ 4 + cos(5) * (-1) * (-1) * (y - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (x - 1) ^ 5 + -sin(5) * (-1) * (-1) * (y - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (x - 1) + -sin(5) * (y - 5) + (-sin(1) * (x - 1) ^ 2 + -cos(5) * (y - 5) ^ 2) / 2! + (-cos(1) * (x - 1) ^ 3 + sin(5) * (y - 5) ^ 3) / 3! + (sin(1) * (x - 1) ^ 4 + cos(5) * (y - 5) ^ 4) / 4! + (cos(1) * (x - 1) ^ 5 + -sin(5) * (y - 5) ^ 5) / 5!
/// ----------------------
/// sin(x) + cos(y)
/// sin(1) + cos(5) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + cos(5) * (-1) * (z_2 - 5) ^ 2) / 2! + (cos(1) * (-1) * (z_1 - 1) ^ 3 + -sin(5) * (-1) * (z_2 - 5) ^ 3) / 3! + (-sin(1) * (-1) * (z_1 - 1) ^ 4 + cos(5) * (-1) * (-1) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (-1) * (-1) * (z_1 - 1) ^ 5 + -sin(5) * (-1) * (-1) * (z_2 - 5) ^ 5) / 5!
/// cos(5) + sin(1) + cos(1) * (z_1 - 1) + -sin(5) * (z_2 - 5) + (-sin(1) * (z_1 - 1) ^ 2 + -cos(5) * (z_2 - 5) ^ 2) / 2! + (-cos(1) * (z_1 - 1) ^ 3 + sin(5) * (z_2 - 5) ^ 3) / 3! + (sin(1) * (z_1 - 1) ^ 4 + cos(5) * (z_2 - 5) ^ 4) / 4! + (cos(1) * (z_1 - 1) ^ 5 + -sin(5) * (z_2 - 5) ^ 5) / 5!
/// ----------------------
/// Received cos(1)
/// Received x + -sin(1) * (y - 1)
/// Received cos(1) * (-1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + -sin(1) * (-1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (-1) * (-1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (-1) * (-1) * (y - 1) ^ 5) / 5!
/// Received -cos(1) * (y - 1) ^ 2 / 2!
/// Received (-x ^ 3 + sin(1) * (y - 1) ^ 3) / 3!
/// Received cos(1) * (y - 1) ^ 4 / 4!
/// Received (x ^ 5 + -sin(1) * (y - 1) ^ 5) / 5!
/// </code>
/// </example>
public static IEnumerable<Entity> TaylorTerms(Entity expr, params (Variable exprVariable, Variable polyVariable, Entity point)[] exprToPolyVars)
Expand Down
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