N

A numeric value of x -- to d significant digits, every one of them right.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageN!↗

N(x)x at the engine's working precision.

SageMathenumeratio_n(x)SymPyN(x, 30)Rustnf(x)

N(x, d)x to exactly d significant digits, the last one correctly rounded.

SageMathenumeratio_n(x, d)SymPyN(x, 30)mpmathenumeratio_n(lambda: (x), d)Juliaenumeratio_nshow(x, d)Rustnf(x)

Typed signatures

Type	Package	Applies
(any, (integer | list<number>)?) -> unknown	analyticreplaces compute-engine	always
(any, (integer | list<number>)?) -> unknown	compute-engine	always

Domain: Arithmetic

Details

• $\mathrm{N}(x, d)$ evaluates x at $d + 20$ digits and again at $d + 10$, rounds both to d, and answers when they agree; when they don't, the value lies near a rounding boundary, and the guard doubles until they do (Ziv's loop). Agreement across precisions is strong evidence rather than a proof.
• Some heads prove their digits instead: Zeta, HurwitzZeta, LerchPhi, PolyLog and StieltjesGamma, on exact real arguments (integers, rationals, decimals). Their kernels bound their own error and return an enclosure, a ball that provably holds the value. When both ends of it round to the same d digits, those digits are certain; when they don't, the working precision rises, with no appeal to agreement. The enclosure travels with the answer (enclosureOf in @enumeratio/analytic). An argument like $\sqrt 2$ arrives already rounded, by an evaluation that bounds nothing, so such a call takes the agreement path.
• The answer carries exactly d significant digits, never more, and the working precision is left as it was: a later evaluation runs at the precision it would have anyway.
• A value exactly halfway between two d-digit answers -- only a terminating decimal, like $1/8$ -- rounds to the even one.
• A list, or a symbolic result, is rounded number by number; integers in it are left alone.
• A head that computes in machine doubles, whatever precision is asked of it, can't be measured more finely: up to 15 digits its double is rounded like any other value; past that it comes back as that double, with the digits it has rather than the d asked for.

Examples

2 more kept as data, checked against the oracles

Basic #

live ↗

In$N(π, 30)$Out$3.141\,592\,653\,589\,793\,238\,462\,643\,383\,28$

The last digit correctly rounded: computing at 30 digits alone gives …059

live ↗

In$N(\sinh(1), 30)$Out$1.175\,201\,193\,643\,801\,456\,882\,381\,850\,6$

A list, number by number

live ↗

In$N(\bigl\lbrack π, e, \sqrt{2}\bigr\rbrack, 5)$Out$\bigl\lbrack3.141\,6, 2.718\,3, 1.414\,2\bigr\rbrack$

An exact rational at the default precision

live ↗

In$N(\frac{1}{7})$Out$0.142\,857\,142\,857\,142\,857\,143$

A fifty-digit approximation of $\frac17$

live ↗

In$N(\frac{1}{7}, 50)$Out$0.142\,857\,142\,857\,142\,857\,142\,857\,142\,857\,142\,857\,142\,857\,142\,857\,14$

$\Gamma(3.3)$ to 50 digits

live ↗

In$N(Γ(\frac{33}{10}), 50)$Out$2.683\,437\,381\,955\,768\,793\,596\,327\,314\,766\,711\,258\,628\,187\,004\,354\,8$

Scope #

$\pi$ to 50 digits

live ↗

In$N(π, 50)$Out$3.141\,592\,653\,589\,793\,238\,462\,643\,383\,279\,502\,884\,197\,169\,399\,375\,1$

Khinchin's constant to 50 digits

live ↗

In$N(\mathrm{Khinchin}, 50)$Out$2.685\,452\,001\,065\,306\,445\,309\,714\,835\,481\,795\,693\,820\,382\,293\,994\,5$

Out of the real domain, the result is complex

live ↗

In$N(\arcsin(3))$Out$1.570\,796\,326\,794\,896\,619\,231\,321\,5-1.762\,747\,174\,039\,086\,050\,47i$

An undefined symbol stays as it is

live ↗

In$N(c)$Out$c$

The requested precision does not change a symbol

live ↗

In$N(c, 47)$Out$c$

A plain symbol is unchanged

live ↗

In$N(\mathrm{poly})$Out$\mathrm{poly}$

Symbolic powers get a float exponent

live ↗

In$N(\sqrt{x}^{3})$Out$x^{1.5}$

A float coefficient inside a symbolic expression

live ↗

In$N(π^{x})$Out$3.141\,592\,653\,589\,793\,238\,462\,643^{x}$

A symbol stays symbolic

live ↗

In$N(\mathrm{int})$Out$\mathrm{int}$

A numeric constant is turned into a float within a symbolic expression

live ↗

In$N(π-2\mathrm{pit}, \bigl\lbrack∞, 20\bigr\rbrack)$Out$3.141\,592\,653\,589\,793\,238\,46-2\mathrm{pit}$

Another symbol is unchanged

live ↗

In$N(\mathrm{crt})$Out$\mathrm{crt}$

Even with a precision it is left alone

live ↗

In$N(\mathrm{crt}, 47)$Out$\mathrm{crt}$

live ↗

In$N(2^{1\,000\,000})$Out$9.900\,656\,229\,295\,898\,250\,7\cdot10^{301029}$

live ↗

In$N(π^{10}, \bigl\lbrack∞, 5\bigr\rbrack)$Out$93\,648.047\,48$

live ↗

In$N(π^{10}, \bigl\lbrack∞, 1\bigr\rbrack)$Out$93\,648$

live ↗

In$N(\frac{2x+π}{e-x}, 20)$Out$\frac{2x+3.141\,592\,653\,589\,793\,238\,5}{2.718\,281\,828\,459\,045\,235\,4-x}$

live ↗

In$N(1-\sqrt{1-1/e^{1\,000}}, 20)$Out$0.0$

live ↗

In$N(k, 25)$Out$k$

live ↗

In$N(e^{x+π})$Out$2.718\,281\,828\,459\,045\,235\,36^{x+3.141\,592\,653\,589\,793\,238\,462\,643}$

live ↗

In$N(k)$Out$k$

live ↗

In$N(1\,000!)$Out$4.023\,872\,600\,770\,937\,735\,44\cdot10^{2567}$

Properties #

live ↗

In$N(1-\varphi+\frac{1}{\varphi}, 20)$Out$-2.0\cdot10^{-340}$

live ↗

In$N(1-\varphi+\frac{1}{\varphi}, \bigl\lbrack∞, 20\bigr\rbrack)$Out$0.0$

live ↗

In$N(e^{100}, \bigl\lbrack∞, -10\bigr\rbrack)$Out$2.688\,117\,141\,816\,135\,448\,412\,625\,551\,580\,014\cdot10^{43}$

live ↗

In$N(10\,000\,000\,000(e^{100}-e^{999\,999\,999\,999/10\,000\,000\,000}), \bigl\lbrack∞, 20\bigr\rbrack)$Out$2.688\,117\,141\,681\,729\,591\,326\,298\,974\,395\,630\,530\,648\,558\,808\,379\,370\,953\,675\,983\,977\cdot10^{43}$

live ↗

In$N(e^{\sin(1\,000)})$Out$2.286\,173\,685\,812\,171\,102\,23$

live ↗

In$N(e^{\sin(1\,000)}, 47)$Out$2.286\,173\,685\,812\,171\,102\,225\,890\,854\,533\,061\,221\,871\,278\,512$

Possible issues #

The accuracy-goal form N(x, [20, PositiveInfinity]) keeps the digits that plain N(x, 20) cancels to 0.0, in the row above

live ↗

In$N(1-\sqrt{1-1/e^{1\,000}}, \bigl\lbrack20, ∞\bigr\rbrack)$Out$2.537\,979\,448\,774\,728\,382\,6\cdot10^{-435}$

An exact tie rounds to the even digit: 0.125 to 0.12

live ↗

In$N(\frac{1}{8}, 2)$Out$0.12$

...and 0.375 to 0.38

live ↗

In$N(\frac{3}{8}, 2)$Out$0.38$

About

Examples	38
Checked against	Wolfram Language 34 agree4 differSageMath 36 agree1 unevaluatedSymPy 31 agree4 differ1 errormpmath 20 agree3 differJulia 20 agree2 differ1 errorRust 9 agree4 differ4 errorOscar 2 differ
Crosswalk	Wolfram LanguageSageMath ×2SymPy ×2mpmathJuliaRust ×2

Implementation

mappedsympypackages/oracle/src/mappings.ts

mpmath and the Julia libraries are numeric already; SymPy and Sage evaluate exactly unless asked.

mappedmpmathpackages/oracle/src/mappings.ts

mpmath and the Julia libraries are numeric already; SymPy and Sage evaluate exactly unless asked.

mappedsagepackages/oracle/src/mappings.ts

mpmath and the Julia libraries are numeric already; SymPy and Sage evaluate exactly unless asked.

mappedoscarpackages/oracle/src/mappings.ts

mpmath and the Julia libraries are numeric already; SymPy and Sage evaluate exactly unless asked.

mappedjuliapackages/oracle/src/mappings.ts

mpmath and the Julia libraries are numeric already; SymPy and Sage evaluate exactly unless asked.

mappedrustpackages/oracle/src/mappings.ts

mpmath and the Julia libraries are numeric already; SymPy and Sage evaluate exactly unless asked.

mappedsympypackages/oracle/src/mappings.ts

The digit count is dropped: lanes compare numbers to a relative tolerance, so the full-precision value is the better witness.

mappedmpmathpackages/oracle/src/mappings.ts

Past a double's 15 digits the operand runs at the requested precision (plus 20 guard digits) and the answer is rounded to it; below, the digit count is dropped.

mappedsagepackages/oracle/src/mappings.ts

The digit count only sets the starting precision, and the value is evaluated until stable: lanes compare numbers to a relative tolerance, so the full-precision value is the better witness.

mappedoscarpackages/oracle/src/mappings.ts

The digit count is dropped: lanes compare numbers to a relative tolerance, so the full-precision value is the better witness.

mappedjuliapackages/oracle/src/mappings.ts

N(x, d) as the answer is written out to d significant digits, ties to the even digit (Wolfram shows N[1/8, 2] as 0.12), so the scan compares digit for digit; below another call it is the rounded number.

mappedrustpackages/oracle/src/mappings.ts

The digit count is dropped: lanes compare numbers to a relative tolerance, so the full-precision value is the better witness.

See also: Round, Interval