devela/num/float/
constants.rs

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1// devela::num::float::constants
2//
3//! Defines `ExtFloatConst` and implements it for floating-point primitives.
4//
5// TOC
6// - trait ExtFloatConst
7// - CONST shared doc strings
8// - macro impl_ext_float_const!
9// - struct TempFloat
10//
11// WAIT: [more_float_constants](https://github.com/rust-lang/rust/issues/103883)
12// - SQRT_3, FRAC_1_SQRT_3, FRAC_1_SQRT_PI, PHI, EGAMMA…
13//
14// NOTE: In sync with num::float:wrapper::consts
15
16#![allow(clippy::excessive_precision)]
17
18use crate::Float; // because of float_technical_const_impls! consts
19
20#[cfg(feature = "nightly_float")]
21use ::core::{f128, f16};
22
23#[doc = crate::TAG_NAMESPACE!()]
24/// Extension trait for floating-point types. Associated constants.
25///
26/// # Constants
27///
28/// - Identities:
29/// [`ONE`], [`ZERO`], [`NEG_ONE`], [`NEG_ZERO`].
30///
31/// - Representation, precision and computational bounds:
32/// [`NAN`], [`INFINITY`], [`NEG_INFINITY`], [`MIN`], [`MIN_POSITIVE`], [`MAX`], [`MIN_EXP`],
33/// [`MAX_EXP`], [`MIN_10_EXP`], [`MAX_10_EXP`], [`EPSILON`], [`RADIX`], [`DIGITS`],
34/// [`MANTISSA_DIGITS`], [`SIGNIFICAND_BITS`], [`EXPONENT_BIAS`], [`EXPONENT_BITS`].
35///
36/// - Arc degrees: [`ARC_DEGREE`], [`ARC_MINUTE`], [`ARC_SECOND`].
37///
38/// - Pi (π) related:
39/// <big>
40/// [π], [π/2], [π/3], [π/4], [π/6], [π/8], [√π], [1/π], [1/√π], [2/π], [2/√π].
41/// </big>
42///
43/// - Tau (τ) related:
44/// <big>
45/// [τ], [τ/2], [τ/3], [τ/4], [τ/5], [τ/6], [τ/8], [τ/9], [τ/12], [τ/16], [τ/24], [τ/72],
46/// [τ/360], [360/τ], [√τ], [1/τ], [1/√τ], [2/τ], [2/√τ].
47/// </big>
48///
49/// - Phi (φ) related:
50/// <big>[φ], [φ²], [1/φ], [-1/φ], [√φ], [1/√φ],</big>
51/// [`TRIBONACCI`].
52///
53/// - Related to integer roots:
54/// <big>
55/// [√2], [1/√2], [√3], [1/√3], [√5], [√6], [√7], [√8], [√10], [√11], [√12], [∛2], [∛3], [1/∛3].
56/// </big>
57///
58/// - Other constants:
59/// [`E`], [`EGAMMA`], [`LOG2_E`], [`LOG2_10`], [`LOG10_E`], [`LOG10_2`], [`LN_2`], [`LN_10`].
60///
61// ---------------------
62/// [`ONE`]: Self::ONE
63/// [`ZERO`]: Self::ZERO
64/// [`NEG_ONE`]: Self::NEG_ONE
65/// [`NEG_ZERO`]: Self::NEG_ZERO
66///
67/// [`NAN`]: Self::NAN
68/// [`INFINITY`]: Self::INFINITY
69/// [`NEG_INFINITY`]: Self::NEG_INFINITY
70/// [`MIN`]: Self::MIN
71/// [`MIN_POSITIVE`]: Self::MIN_POSITIVE
72/// [`MAX`]: Self::MAX
73/// [`MIN_EXP`]: Self::MIN_EXP
74/// [`MAX_EXP`]: Self::MAX_EXP
75/// [`MIN_10_EXP`]: Self::MIN_10_EXP
76/// [`MAX_10_EXP`]: Self::MAX_10_EXP
77/// [`EPSILON`]: Self::EPSILON
78/// [`RADIX`]: Self::RADIX
79/// [`DIGITS`]: Self::DIGITS
80/// [`MANTISSA_DIGITS`]: Self::MANTISSA_DIGITS
81/// [`SIGNIFICAND_BITS`]: Self::SIGNIFICAND_BITS
82/// [`EXPONENT_BIAS`]: Self::EXPONENT_BIAS
83/// [`EXPONENT_BITS`]: Self::EXPONENT_BITS
84///
85/// [`ARC_DEGREE`]: Self::ARC_DEGREE
86/// [`ARC_MINUTE`]: Self::ARC_MINUTE
87/// [`ARC_SECOND`]: Self::ARC_SECOND
88///
89/// [π]: Self::PI
90/// [π/2]: Self::FRAC_PI_2
91/// [π/3]: Self::FRAC_PI_3
92/// [π/4]: Self::FRAC_PI_4
93/// [π/6]: Self::FRAC_PI_6
94/// [π/8]: Self::FRAC_PI_8
95/// [√π]: Self::SQRT_PI
96/// [1/π]: Self::FRAC_1_PI
97/// [1/√π]: Self::FRAC_1_SQRT_PI
98/// [1/√2π]: Self::FRAC_1_SQRT_2PI
99/// [2/π]: Self::FRAC_2_PI
100/// [2/√π]: Self::FRAC_2_SQRT_PI
101///
102/// [τ]: Self::TAU
103/// [τ/2]: Self::FRAC_TAU_2
104/// [τ/3]: Self::FRAC_TAU_3
105/// [τ/4]: Self::FRAC_TAU_4
106/// [τ/5]: Self::FRAC_TAU_5
107/// [τ/6]: Self::FRAC_TAU_6
108/// [τ/8]: Self::FRAC_TAU_8
109/// [τ/9]: Self::FRAC_TAU_9
110/// [τ/12]: Self::FRAC_TAU_12
111/// [τ/16]: Self::FRAC_TAU_16
112/// [τ/24]: Self::FRAC_TAU_24
113/// [τ/72]: Self::FRAC_TAU_72
114/// [τ/360]: Self::FRAC_TAU_360
115/// [360/τ]: Self::FRAC_360_TAU
116/// [√τ]: Self::SQRT_TAU
117/// [1/τ]: Self::FRAC_1_TAU
118/// [1/√τ]: Self::FRAC_1_SQRT_TAU
119/// [2/τ]: Self::FRAC_2_TAU
120/// [2/√τ]: Self::FRAC_2_SQRT_TAU
121///
122/// [φ]: Self::PHI
123/// [φ²]: Self::SQ_PHI
124/// [1/φ]: Self::FRAC_1_PHI
125/// [-1/φ]: Self::NEG_FRAC_1_PHI
126/// [√φ]: Self::SQRT_PHI
127/// [1/√φ]: Self::FRAC_1_SQRT_PHI
128/// [`TRIBONACCI`]: Self::TRIBONACCI
129///
130/// [√2]: Self::SQRT_2
131/// [1/√2]: Self::FRAC_1_SQRT_2
132/// [√3]: Self::SQRT_3
133/// [1/√3]: Self::FRAC_1_SQRT_3
134/// [√5]: Self::SQRT_5
135/// [√6]: Self::SQRT_6
136/// [√7]: Self::SQRT_7
137/// [√8]: Self::SQRT_8
138/// [√10]: Self::SQRT_10
139/// [√11]: Self::SQRT_11
140/// [√12]: Self::SQRT_12
141/// [∛2]: Self::CBRT_2
142/// [∛3]: Self::CBRT_3
143/// [1/∛3]: Self::FRAC_1_CBRT_3
144///
145/// [`E`]: Self::E
146/// [`EGAMMA`]: Self::EGAMMA
147/// [`LOG2_E`]: Self::LOG2_E
148/// [`LOG2_10`]: Self::LOG2_10
149/// [`LOG10_E`]: Self::LOG10_E
150/// [`LOG10_2`]: Self::LOG10_2
151/// [`LN_2`]: Self::LN_2
152/// [`LN_10`]: Self::LN_10
153#[rustfmt::skip]
154pub trait ExtFloatConst: Sized {
155    // identities
156    #[doc = ONE!()]                 const ONE: Self;
157    #[doc = ZERO!()]                const ZERO: Self;
158    #[doc = NEG_ONE!()]             const NEG_ONE: Self;
159    #[doc = NEG_ZERO!()]            const NEG_ZERO: Self;
160    // representation, precision and range
161    #[doc = NAN!()]                 const NAN: Self;
162    #[doc = INFINITY!()]            const INFINITY: Self;
163    #[doc = NEG_INFINITY!()]        const NEG_INFINITY: Self;
164    #[doc = MIN!()]                 const MIN: Self;
165    #[doc = MIN_POSITIVE!()]        const MIN_POSITIVE: Self;
166    #[doc = MAX!()]                 const MAX: Self;
167    #[doc = MIN_EXP!()]             const MIN_EXP: i32;
168    #[doc = MAX_EXP!()]             const MAX_EXP: i32;
169    #[doc = MIN_10_EXP!()]          const MIN_10_EXP: i32;
170    #[doc = MAX_10_EXP!()]          const MAX_10_EXP: i32;
171    #[doc = EPSILON!()]             const EPSILON: Self;
172    #[doc = LOW_MARGIN!()]          const LOW_MARGIN: Self;
173    #[doc = MEDIUM_MARGIN!()]       const MEDIUM_MARGIN: Self;
174    #[doc = HIGH_MARGIN!()]         const HIGH_MARGIN: Self;
175    #[doc = RADIX!()]               const RADIX: u32;
176    #[doc = DIGITS!()]              const DIGITS: u32;
177    #[doc = MANTISSA_DIGITS!()]     const MANTISSA_DIGITS: u32;
178    #[doc = SIGNIFICAND_BITS!()]    const SIGNIFICAND_BITS: u32;
179    #[doc = EXPONENT_BIAS!()]       const EXPONENT_BIAS: u32;
180    #[doc = EXPONENT_BITS!()]       const EXPONENT_BITS: u32;
181    // pi
182    #[doc = PI!()]                  const PI: Self;
183    #[doc = FRAC_PI_2!()]           const FRAC_PI_2: Self;
184    #[doc = FRAC_PI_3!()]           const FRAC_PI_3: Self;
185    #[doc = FRAC_PI_4!()]           const FRAC_PI_4: Self;
186    #[doc = FRAC_PI_6!()]           const FRAC_PI_6: Self;
187    #[doc = FRAC_PI_8!()]           const FRAC_PI_8: Self;
188    #[doc = SQRT_PI!()]             const SQRT_PI: Self;
189    #[doc = FRAC_1_PI!()]           const FRAC_1_PI: Self;
190    #[doc = FRAC_1_SQRT_PI!()]      const FRAC_1_SQRT_PI: Self;
191    #[doc = FRAC_1_SQRT_2PI!()]     const FRAC_1_SQRT_2PI: Self;
192    #[doc = FRAC_2_PI!()]           const FRAC_2_PI: Self;
193    #[doc = FRAC_2_SQRT_PI!()]      const FRAC_2_SQRT_PI: Self;
194    // tau
195    #[doc = TAU!()]                 const TAU: Self;
196    #[doc = FRAC_TAU_2!()]          const FRAC_TAU_2: Self;
197    #[doc = FRAC_TAU_3!()]          const FRAC_TAU_3: Self;
198    #[doc = FRAC_TAU_4!()]          const FRAC_TAU_4: Self;
199    #[doc = FRAC_TAU_5!()]          const FRAC_TAU_5: Self;
200    #[doc = FRAC_TAU_6!()]          const FRAC_TAU_6: Self;
201    #[doc = FRAC_TAU_8!()]          const FRAC_TAU_8: Self;
202    #[doc = FRAC_TAU_9!()]          const FRAC_TAU_9: Self;
203    #[doc = FRAC_TAU_12!()]         const FRAC_TAU_12: Self;
204    #[doc = FRAC_TAU_16!()]         const FRAC_TAU_16: Self;
205    #[doc = FRAC_TAU_24!()]         const FRAC_TAU_24: Self;
206    #[doc = FRAC_TAU_72!()]         const FRAC_TAU_72: Self;
207    #[doc = FRAC_TAU_360!()]        const FRAC_TAU_360: Self;
208    #[doc = FRAC_360_TAU!()]        const FRAC_360_TAU: Self;
209    #[doc = SQRT_TAU!()]            const SQRT_TAU: Self;
210    #[doc = FRAC_1_TAU!()]          const FRAC_1_TAU: Self;
211    #[doc = FRAC_1_SQRT_TAU!()]     const FRAC_1_SQRT_TAU: Self;
212    #[doc = FRAC_2_TAU!()]          const FRAC_2_TAU: Self;
213    #[doc = FRAC_2_SQRT_TAU!()]     const FRAC_2_SQRT_TAU: Self;
214    // arc degrees
215    #[doc = ARC_DEGREE!()]          const ARC_DEGREE: Self;
216    #[doc = ARC_MINUTE!()]          const ARC_MINUTE: Self;
217    #[doc = ARC_SECOND!()]          const ARC_SECOND: Self;
218    // phi
219    #[doc = PHI!()]                 const PHI: Self;
220    #[doc = SQ_PHI!()]              const SQ_PHI: Self;
221    #[doc = FRAC_1_PHI!()]          const FRAC_1_PHI: Self;
222    #[doc = NEG_FRAC_1_PHI!()]      const NEG_FRAC_1_PHI: Self;
223    #[doc = SQRT_PHI!()]            const SQRT_PHI: Self;
224    #[doc = FRAC_1_SQRT_PHI!()]     const FRAC_1_SQRT_PHI: Self;
225    #[doc = TRIBONACCI!()]          const TRIBONACCI: Self;
226    // sqrt
227    #[doc = SQRT_2!()]              const SQRT_2: Self;
228    #[doc = FRAC_1_SQRT_2!()]       const FRAC_1_SQRT_2: Self;
229    #[doc = SQRT_3!()]              const SQRT_3: Self;
230    #[doc = FRAC_1_SQRT_3!()]       const FRAC_1_SQRT_3: Self;
231    #[doc = SQRT_5!()]              const SQRT_5: Self;
232    #[doc = SQRT_6!()]              const SQRT_6: Self;
233    #[doc = SQRT_7!()]              const SQRT_7: Self;
234    #[doc = SQRT_8!()]              const SQRT_8: Self;
235    #[doc = SQRT_10!()]             const SQRT_10: Self;
236    #[doc = SQRT_11!()]             const SQRT_11: Self;
237    #[doc = SQRT_12!()]             const SQRT_12: Self;
238    #[doc = CBRT_2!()]              const CBRT_2: Self;
239    #[doc = CBRT_3!()]              const CBRT_3: Self;
240    #[doc = FRAC_1_CBRT_3!()]       const FRAC_1_CBRT_3: Self;
241    // other
242    #[doc = E!()]                   const E: Self;
243    #[doc = EGAMMA!()]              const EGAMMA: Self;
244    #[doc = LOG2_E!()]              const LOG2_E: Self;
245    #[doc = LOG2_10!()]             const LOG2_10: Self;
246    #[doc = LOG10_E!()]             const LOG10_E: Self;
247    #[doc = LOG10_2!()]             const LOG10_2: Self;
248    #[doc = LN_2!()]                const LN_2: Self;
249    #[doc = LN_10!()]               const LN_10: Self;
250}
251
252// Define shared doc strings
253crate::CONST! { pub(in crate::num::float),
254    // identities
255    ONE = r#"The multiplicative identity 1."#;
256    ZERO = r#"The additive identity 0."#;
257    NEG_ONE = r#"The negative of the multiplicative identity -1."#;
258    NEG_ZERO = r#"The negative of the additive identity -0."#;
259
260    // representation, precision and range
261    NAN = r#"Not a Number (NaN)."#;
262    INFINITY = r#"Infinity (∞)."#;
263    NEG_INFINITY = r#"Negative infinity (-∞)."#;
264    MIN = r#"Smallest finite value."#;
265    MIN_POSITIVE = r#"Smallest positive normal value."#;
266    MAX = r#"Largest finite value."#;
267    MIN_EXP = r#"One greater than the minimum possible normal power of 2 exponent."#;
268    MAX_EXP = r#"Maximum possible power of 2 exponent."#;
269    MIN_10_EXP = r#"Minimum *x* for which 10<sup>*x*</sup> is normal."#;
270    MAX_10_EXP = r#"Maximum *x* for which 10<sup>*x*</sup> is normal."#;
271    EPSILON = r#"Machine epsilon value.
272    <p>This is the smallest difference detectable between 1.0 and the next
273    representable number in the floating-point format.</p>"#;
274    LOW_MARGIN = r#"Allows for minimal deviation; use for high precision needs.."#;
275    MEDIUM_MARGIN = r#"Accommodates moderate deviation; balances precision and flexibility."#;
276    HIGH_MARGIN = r#"Permits generous deviation; suitable for less precise scenarios."#;
277    RADIX = r#"The radix or base of the internal representation."#;
278    DIGITS = r#"Approximate number of significant digits in base 10."#;
279    MANTISSA_DIGITS = r#"Number of significant digits in base 2."#;
280    SIGNIFICAND_BITS = "Number of explicit bits used to represent the significand (or mantissa).";
281    EXPONENT_BIAS = r#"Exponent bias for representing both positive and negative exponents."#;
282    EXPONENT_BITS = r#"Number of bits used to represent the exponent."#;
283
284    // pi
285    PI = r#"$ π = \frac{1}{2} τ = 180º $
286    ([A000796](https://oeis.org/A000796/constant))
287    `≈ 3.14159265…`
288    <p>*The ratio of the circumference to the diameter, a half-turn*.</p>"#;
289    FRAC_PI_2 = r#"$ π/2 = τ/4 = 90º $
290    ([A019669](https://oeis.org/A019669/constant))
291    `≈ 1.57079632…`"#;
292    FRAC_PI_3 = r#"$ π/3 = τ/6 = 60º $
293    ([A019670](https://oeis.org/A019670/constant))
294    `≈ 1.04719755…`"#;
295    FRAC_PI_4 = r#"$ π/4 = τ/8 = 45º $
296    ([A003881](https://oeis.org/A003881/constant))
297    `≈ 0.78539816…`"#;
298    FRAC_PI_6 = r#"$ π/6 = τ/12 = 30º $
299    ([A019673](https://oeis.org/A019673/constant))
300    `≈ 0.52359877…`"#;
301    FRAC_PI_8 = r#"$ π/8 = τ/16 = 22.5º $
302    ([A019675](https://oeis.org/A019675/constant))
303    `≈ 0.39269908…`"#;
304    SQRT_PI = r#"$ \sqrt{π} = \sqrt{\frac{1}{2} τ} $
305    ([A002161](https://oeis.org/A002161/constant))
306    `≈ 1.77245385…`"#;
307    FRAC_1_PI = r#"$ 1/π = 2/τ $
308    ([A049541](https://oeis.org/A049541/constant))
309    `≈ 0.31830988…`"#;
310    FRAC_1_SQRT_PI = r#"$ 1/\sqrt{π} = 1/\sqrt{τ/2} $
311    ([A087197](https://oeis.org/A087197/constant))
312    `≈ 0.56418958…`"#;
313    FRAC_1_SQRT_2PI = r#"$ 1/\sqrt{2π} = 1/\sqrt{τ} $
314    ([A231863](https://oeis.org/A231863/constant))
315    `≈ 0.39894228…`"#;
316    FRAC_2_PI = r#"$ 2/π $
317    ([A060294](https://oeis.org/A060294/constant))
318    `≈ 0.63661977…`
319    <p>*Buffon's constant*.</p>"#;
320    FRAC_2_SQRT_PI = r#"$ 2/\sqrt{π} $
321    ([A190732](https://oeis.org/A190732/constant))
322    `≈ 1.12837916…`"#;
323
324    // tau
325    TAU = r#"$ τ = 2π = 360º $
326    ([A019692](https://oeis.org/A019692/constant))
327    `≈ 6.28318530…`
328    <p>*The ratio of the circumference to the radius, a full-turn*.</p>"#;
329    FRAC_TAU_2 = r#"$ τ/2 = π = 180º $
330    ([A000796](https://oeis.org/A000796/constant))
331    `≈ 3.14159265…`"#;
332    FRAC_TAU_3 = r#"$ τ/3  = 2π/3 = 120º $
333    ([A019693](https://oeis.org/A019693/constant))
334    `≈ 2.09439510…`"#;
335    FRAC_TAU_4 = r#"$ τ/4 = π/2 = 90º $
336    ([A019693](https://oeis.org/A019693/constant))
337    `≈ 1.57079632…`"#;
338    FRAC_TAU_5 = r#"$ τ/5 = 2π/5 = 72º $
339    ([A019694](https://oeis.org/A019694/constant))
340    `≈ 1.25663706…`"#;
341    FRAC_TAU_6 = r#"$ τ/6 = π/3 = 60º $
342    ([A019670](https://oeis.org/A019670/constant))
343    `≈ 1.04719755…`"#;
344    FRAC_TAU_8 = r#"$ τ/8 = π/4 = 45º $
345    ([A003881](https://oeis.org/A003881/constant))
346    `≈ 0.78539816…`"#;
347    FRAC_TAU_9 = r#"$ τ/9 = 2π/9 = 40º $
348    ([A019696](https://oeis.org/A019696/constant))
349    `≈ 0.69813170…`"#;
350    FRAC_TAU_12 = r#"$ τ/12 = π/6 = 30º $
351    ([A019673](https://oeis.org/A019673/constant))
352    `≈ 0.52359877…`"#;
353    FRAC_TAU_16 = r#"$ τ/16 = π/8 = 22.5º $
354    ([A019675](https://oeis.org/A019675/constant))
355    `≈ 0.39269908…`"#;
356    FRAC_TAU_24 = r#"$ τ/24 = π/12 = 15º $
357    ([A019679](https://oeis.org/A019679/constant))
358    `≈ 0.26179938…`"#;
359    FRAC_TAU_72 = r#"$ τ/72 = π/36 = 5º $
360    `≈ 0.08726646…`"#;
361    FRAC_TAU_360 = r#"$ τ/360 = π/180 = 1º $ *arc degree*
362    ([A019685](https://oeis.org/A019685),
363    [wikipedia](https://en.wikipedia.org/wiki/Degree_(angle)))
364    `≈ 0.01745329…`"#;
365    FRAC_360_TAU = r#"$ 360/τ = 180/π $
366    ([A072097](https://oeis.org/A072097/constant))
367    `≈ 57.2957795…`"#;
368    SQRT_TAU = r#"$ \sqrt{τ} = \sqrt{2π} $
369    ([A019727](https://oeis.org/A019727/constant))
370    `≈ 2.50662827…`"#;
371    FRAC_1_TAU = r#"$ 1/τ = 1/2π $
372    ([A086201](https://oeis.org/A086201/constant))
373    `≈ 0.15915494…`"#;
374    FRAC_1_SQRT_TAU = r#"$ 1/\sqrt{τ} = 1/\sqrt{2π} $
375    ([A231863](https://oeis.org/A231863/constant))
376    `≈ 0.39894228…`"#;
377    FRAC_2_TAU = r#"$ 2/τ = 1/π $
378    ([A049541](https://oeis.org/A049541/constant))
379    `≈ 0.31830988…`"#;
380    FRAC_2_SQRT_TAU = r#"$ 2/\sqrt{τ} = \sqrt{2/π} $
381    ([A076668](https://oeis.org/A076668/constant))
382    `≈ 0.79788456…`"#;
383
384    // arc degrees
385    ARC_DEGREE = r#"$ τ/360 = π/180 = 1º $ *arc degree*
386    ([A019685](https://oeis.org/A019685),
387    [wikipedia](https://en.wikipedia.org/wiki/Degree_(angle)))
388    `≈ 0.01745329…`"#;
389    ARC_MINUTE = r#"$ τ/(360*60) = 1' $ *arc minute*
390    ([wikipedia](https://en.wikipedia.org/wiki/Minute_and_second_of_arc))
391    `≈ 0.00029088…`"#;
392    ARC_SECOND = r#"$ τ/(360 * 60 * 60) = 1'' $ *arc second*
393    ([wikipedia](https://en.wikipedia.org/wiki/Minute_and_second_of_arc))
394    `≈ 0.00000484…`"#;
395
396    // phi
397    PHI = r#"$ φ  = (1+\sqrt{5})/2 $
398    ([A001622](https://oeis.org/A001622/constant))
399    `≈ 1.61803398…`
400    <p>*The golden ratio*.</p>
401    <p>Continued fraction: $ [1;1,1,1,…] $</p>"#;
402    SQ_PHI = r#"$ φ^2 = φ+1 = (3+\sqrt{5})/2 $
403    ([A104457](https://oeis.org/A104457/constant))
404    `≈ 2.61803398…`"#;
405    FRAC_1_PHI = r#"$ 1/φ = φ-1 $
406    ([A094214](https://oeis.org/A094214/constant))
407    `≈ 0.61803398…`
408    <p>*The reciprocal of [φ][Self#PHI]*.</p>"#;
409    NEG_FRAC_1_PHI = r#"$ -1/φ = 1-φ $
410    `≈ -0.61803398…`
411    <p>*The negative reciprocal of [φ][Self#PHI] and its conjugate in $ x^2-x-1 $*.</p>"#;
412    SQRT_PHI = r#"$ \sqrt{φ} $
413    ([A139339](https://oeis.org/A139339/constant))
414    `≈ 1.27201964…`"#;
415    FRAC_1_SQRT_PHI = r#"$ 1/\sqrt{φ} = \sqrt{φ/φ^2} = \sqrt{φ^2-2} $
416    ([A197762](https://oeis.org/A197762/constant))
417    `≈ 0.78615137…`"#;
418    TRIBONACCI = r#"([A058265](https://oeis.org/A058265/constant))
419    `≈ 1.83928675…`
420    <p>*The tribonacci constant*.</p>"#;
421
422    // integer roots
423    SQRT_2 = r#"$ \sqrt{2} $
424    ([A002193](https://oeis.org/A002193/constant),
425    [wikipedia](https://en.wikipedia.org/wiki/Square_root_of_2))
426    `≈ 1.41421356…`"#;
427    FRAC_1_SQRT_2 = r#"$ 1/\sqrt{2} = \sqrt{1/2} $
428    ([A010503](https://oeis.org/A010503/constant),
429    [wikipedia](https://en.wikipedia.org/wiki/Square_root_of_2#Multiplicative_inverse))
430    `≈ 0.70710678…`"#;
431    SQRT_3 = r#"$ \sqrt{3} $
432    ([A002194](https://oeis.org/A002194/constant),
433    [wikipedia](https://en.wikipedia.org/wiki/Square_root_of_3))
434    `≈ 1.73205080…`"#;
435    FRAC_1_SQRT_3 = r#"$ 1/\sqrt{3} = \sqrt{1/3} $
436    ([A020760](https://oeis.org/A002194/constant),
437    `≈ 0.57735026…`"#;
438    SQRT_5 = r#"$ \sqrt{5} $
439    ([A002163](https://oeis.org/A002163/constant),
440    [wikipedia](https://en.wikipedia.org/wiki/Square_root_of_5))
441    `≈ 2.23606797…`"#;
442    SQRT_6 = r#"$ \sqrt{6} $
443    ([A010464](https://oeis.org/A010464/constant))
444    `≈ 2.44948974…`"#;
445    SQRT_7 = r#"$ \sqrt{7} $
446    ([A010465](https://oeis.org/A010465/constant))
447    `≈ 2.64575131…`"#;
448    SQRT_8 = r#"$ \sqrt{8} $
449    ([A010466](https://oeis.org/A010466/constant))
450    `≈ 2.82842712…`"#;
451    SQRT_10 = r#"$ \sqrt{10} $
452    ([A010467](https://oeis.org/A010467/constant))
453    `≈ 3.16227766…`"#;
454    SQRT_11 = r#"$ \sqrt{11} $
455    ([A010468](https://oeis.org/A010468/constant))
456    `≈ 3.31662479…`"#;
457    SQRT_12 = r#"$ \sqrt{12} $
458    ([A010469](https://oeis.org/A010469/constant))
459    `≈ 3.46410161…`"#;
460    CBRT_2 = r#"$ \sqrt[\small 3]{2} $
461    ([A002580](https://oeis.org/A002580/constant),
462    [wikipedia](https://en.wikipedia.org/wiki/Doubling_the_cube))
463    `≈ 1.25992104…`"#;
464    CBRT_3 = r#"$ \sqrt[\small 3]{3} $
465    ([A002581](https://oeis.org/A002581/constant))
466    `≈ 1.44224957…`"#;
467    FRAC_1_CBRT_3 = r#"$ 1/\sqrt[\small 3]{3} = (\normalsize\frac{1}{3})^{\small\frac{1}{3}} $
468    ([A072365](https://oeis.org/A072365/constant))
469    `≈ 0.69336127…`"#;
470
471    // other
472    E = r#"$ e $
473    ([A001113](https://oeis.org/A001113/constant))
474    `≈ 2.71828182…`
475    <p>*The Euler number or Napier's constant*.</p>
476    <p>Continuous fraction: $ [2;1,2,1,1,4,1,1,6,1,…,1,2n,1,…] $</p>"#;
477    EGAMMA = r#"$ γ $
478    ([A001620](https://oeis.org/A001620/constant))
479    `≈ 0.57721566…`
480    <p>*Gamma, or the Euler-Mascheroni constant.*</p>"#;
481    LOG2_E = r#"$ \log_2{e} $
482    ([A007525](https://oeis.org/A007525/constant))
483    `≈ 1.44269504…`"#;
484    LOG2_10 = r#"log<sub>2</sub>(10)
485    ([A020862](https://oeis.org/A020862/constant))
486    `≈ 3.32192809…`"#;
487    LOG10_E = r#"log<sub>10</sub>(e)
488    ([A002285](https://oeis.org/A002285/constant))
489    `≈ 0.43429448…`"#;
490    LOG10_2 = r#"log<sub>10</sub>(2)
491    ([A007524](https://oeis.org/A007524/constant))
492    `≈ 0.30102999…`"#;
493    LN_2 = r#"ln(2)
494    ([A002162](https://oeis.org/A002162/constant))
495    `≈ 0.69314718…`"#;
496    LN_10 = r#"ln(10)
497    ([A002392](https://oeis.org/A002392/constant))
498    `≈ 2.30258509…`"#;
499}
500
501/// impl mathematical constants
502///
503/// $f: the floating-point type.
504macro_rules! impl_ext_float_const {
505    ($( $(#[$attrs:meta])* $f:ty ),+) => { $( impl_ext_float_const![@$(#[$attrs])* $f]; )+ };
506    (@$(#[$attrs:meta])* $f:ty) => {
507        /// # *Mathematical constants*.
508        $(#[$attrs])*
509        impl ExtFloatConst for $f {
510            // identities
511            const ONE: $f = 1.0;
512            const ZERO: $f = 0.0;
513            const NEG_ONE: $f = -1.0;
514            const NEG_ZERO: $f = -0.0;
515
516            // representation, precision and range
517            const NAN: $f = <$f>::NAN;
518            const INFINITY: $f = <$f>::INFINITY;
519            const NEG_INFINITY: $f = <$f>::NEG_INFINITY;
520            const EPSILON: $f = <$f>::EPSILON;
521            const LOW_MARGIN: $f = TempFloat::<$f>::LOW_MARGIN;
522            const MEDIUM_MARGIN: $f = TempFloat::<$f>::MEDIUM_MARGIN;
523            const HIGH_MARGIN: $f = TempFloat::<$f>::HIGH_MARGIN;
524            const RADIX: u32 = <$f>::RADIX;
525            const DIGITS: u32 = <$f>::DIGITS;
526            const MANTISSA_DIGITS: u32 = <$f>::MANTISSA_DIGITS;
527            const SIGNIFICAND_BITS: u32 = Float::<$f>::SIGNIFICAND_BITS;
528            const EXPONENT_BIAS: u32 = Float::<$f>::EXPONENT_BIAS;
529            const EXPONENT_BITS: u32 = Float::<$f>::EXPONENT_BITS;
530            const MIN: $f = <$f>::MIN;
531            const MIN_POSITIVE: $f = <$f>::MIN_POSITIVE;
532            const MAX: $f = <$f>::MAX;
533            const MIN_EXP: i32 = <$f>::MIN_EXP;
534            const MAX_EXP: i32 = <$f>::MAX_EXP;
535            const MIN_10_EXP: i32 = <$f>::MIN_10_EXP;
536            const MAX_10_EXP: i32 = <$f>::MAX_10_EXP;
537
538            // pi
539            const PI: $f =
540             3.14159265358979323846264338327950288419716939937510582097494459230781640628620899;
541            const FRAC_PI_2: $f =
542             1.57079632679489661923132169163975144209858469968755291048747229615390820314310449;
543            const FRAC_PI_3: $f =
544             1.04719755119659774615421446109316762806572313312503527365831486410260546876206966;
545            const FRAC_PI_4: $f =
546             0.78539816339744830961566084581987572104929234984377645524373614807695410157155224;
547            const FRAC_PI_6: $f =
548             0.52359877559829887307710723054658381403286156656251763682915743205130273438103483;
549            const FRAC_PI_8: $f =
550             0.39269908169872415480783042290993786052464617492188822762186807403847705078577612;
551            const SQRT_PI: $f =
552             1.77245385090551602729816748334114518279754945612238712821380778985291128459103218;
553            const FRAC_1_PI: $f =
554             0.31830988618379067153776752674502872406891929148091289749533468811779359526845307;
555            const FRAC_1_SQRT_PI: $f =
556             0.56418958354775628694807945156077258584405062932899885684408572171064246844149341;
557            const FRAC_1_SQRT_2PI: $f = Self::FRAC_1_SQRT_TAU;
558            const FRAC_2_PI: $f =
559             0.63661977236758134307553505349005744813783858296182579499066937623558719053690614;
560            const FRAC_2_SQRT_PI: $f =
561             1.12837916709551257389615890312154517168810125865799771368817144342128493688298683;
562
563            // tau
564            const TAU: $f =
565             6.28318530717958647692528676655900576839433879875021164194988918461563281257241799;
566            const FRAC_TAU_2: $f = Self::PI;
567            const FRAC_TAU_3: $f =
568             2.09439510239319549230842892218633525613144626625007054731662972820521093752413933;
569            const FRAC_TAU_4: $f = Self::FRAC_PI_2;
570            const FRAC_TAU_5: $f =
571             1.25663706143591729538505735331180115367886775975004232838997783692312656251448359;
572            const FRAC_TAU_6: $f = Self::FRAC_PI_3;
573            const FRAC_TAU_8: $f = Self::FRAC_PI_4;
574            const FRAC_TAU_9: $f =
575             0.69813170079773183076947630739544508537714875541669018243887657606840364584137977;
576            const FRAC_TAU_12: $f = Self::FRAC_PI_6;
577            const FRAC_TAU_16: $f = Self::FRAC_PI_8;
578            const FRAC_TAU_24: $f =
579             0.26179938779914943653855361527329190701643078328125881841457871602565136719051741;
580            const FRAC_TAU_72: $f =
581             0.08726646259971647884618453842443063567214359442708627280485957200855045573017247;
582            const FRAC_TAU_360: $f =
583             0.01745329251994329576923690768488612713442871888541725456097191440171009114603449;
584            const FRAC_360_TAU: $f =
585            57.29577951308232087679815481410517033240547246656432154916024386120284714832155263;
586            const SQRT_TAU: $f =
587             2.50662827463100050241576528481104525300698674060993831662992357634229365460784197;
588            const FRAC_1_TAU: $f =
589             0.15915494309189533576888376337251436203445964574045644874766734405889679763422653;
590            const FRAC_1_SQRT_TAU: $f =
591             0.39894228040143267793994605993438186847585863116493465766592582967065792589930183;
592            const FRAC_2_TAU: $f = Self::FRAC_1_PI;
593            const FRAC_2_SQRT_TAU: $f =
594             0.79788456080286535587989211986876373695171726232986931533185165934131585179860367;
595
596            // arc degrees
597            const ARC_DEGREE: $f = Self::FRAC_TAU_360;
598            const ARC_MINUTE: $f =
599             0.00029088820866572159615394846141476878557381198142362090934953190669516818576724;
600            const ARC_SECOND: $f =
601             0.00000484813681109535993589914102357947975956353302372701515582553177825280309612;
602
603            // phi
604            const PHI: $f =
605             1.61803398874989484820458683436563811772030917980576286213544862270526046281890244;
606            const SQ_PHI: $f =
607             2.61803398874989484820458683436563811772030917980576286213544862270526046281890244;
608            const FRAC_1_PHI: $f =
609             0.61803398874989484820458683436563811772030917980576286213544862270526046281890244;
610            const NEG_FRAC_1_PHI: $f =
611            -0.61803398874989484820458683436563811772030917980576286213544862270526046281890244;
612            const SQRT_PHI: $f =
613             1.27201964951406896425242246173749149171560804184009624861664038253929757553606801;
614            const FRAC_1_SQRT_PHI: $f =
615             0.78615137775742328606955858584295892952312205783772323766490197010118204762231091;
616            const TRIBONACCI: $f =
617             1.83928675521416113255185256465328660042417874609759224677875863940420322208196642;
618
619            // integer roots
620            const SQRT_2: $f =
621             1.41421356237309504880168872420969807856967187537694807317667973799073247846210703;
622            const FRAC_1_SQRT_2: $f =
623             0.70710678118654752440084436210484903928483593768847403658833986899536623923105351;
624            const SQRT_3: $f =
625             1.73205080756887729352744634150587236694280525381038062805580697945193301690880003;
626            const FRAC_1_SQRT_3: $f =
627             0.57735026918962576450914878050195745564760175127012687601860232648397767230293334;
628            const SQRT_5: $f =
629             2.23606797749978969640917366873127623544061835961152572427089724541052092563780489;
630            const SQRT_6: $f =
631             2.44948974278317809819728407470589139196594748065667012843269256725096037745731502;
632            const SQRT_7: $f =
633             2.64575131106459059050161575363926042571025918308245018036833445920106882323028362;
634            const SQRT_8: $f =
635             2.82842712474619009760337744841939615713934375075389614635335947598146495692421407;
636            const SQRT_10: $f =
637             3.16227766016837933199889354443271853371955513932521682685750485279259443863923822;
638            const SQRT_11: $f =
639             3.31662479035539984911493273667068668392708854558935359705868214611648464260904384;
640            const SQRT_12: $f =
641             3.46410161513775458705489268301174473388561050762076125611161395890386603381760007;
642            const CBRT_2: $f =
643             1.25992104989487316476721060727822835057025146470150798008197511215529967651395948;
644            const CBRT_3: $f =
645             1.44224957030740838232163831078010958839186925349935057754641619454168759682999733;
646            const FRAC_1_CBRT_3: $f =
647             0.69336127435063470484335227478596179544593511345775403656586369340003543713242292;
648
649            // other
650            const E: $f =
651             2.71828182845904523536028747135266249775724709369995957496696762772407663035354759;
652            const EGAMMA: $f =
653             0.57721566490153286060651209008240243104215933593992359880576723488486772677766467;
654            const LOG2_E: $f =
655             1.44269504088896340735992468100189213742664595415298593413544940693110921918118507;
656            const LOG2_10: $f =
657             3.32192809488736234787031942948939017586483139302458061205475639581593477660862521;
658            const LOG10_E: $f =
659             0.43429448190325182765112891891660508229439700580366656611445378316586464920887077;
660            const LOG10_2: $f =
661             0.30102999566398119521373889472449302676818988146210854131042746112710818927442450;
662            const LN_2: $f =
663             0.69314718055994530941723212145817656807550013436025525412068000949339362196969471;
664            const LN_10: $f =
665             2.30258509299404568401799145468436420760110148862877297603332790096757260967735248;
666        }
667    };
668}
669impl_ext_float_const![f32, f64];
670#[cfg(feature = "nightly_float")]
671impl_ext_float_const![
672    #[cfg_attr(feature = "nightly_doc", doc(cfg(feature = "nightly_float")))]
673    f16,
674    #[cfg_attr(feature = "nightly_doc", doc(cfg(feature = "nightly_float")))]
675    f128
676];
677
678/// Private helper struct to define manual, type-dependendent constants.
679struct TempFloat<T> {
680    _marker: ::core::marker::PhantomData<T>,
681}
682macro_rules! impl_temp_float {
683    () => {
684        #[cfg(feature = "nightly_float")]
685        impl_temp_float![f16: 1e-4, 1e-3, 1e-2]; // ~3–4 decimal digits of precision.
686        impl_temp_float![f32: 1e-7, 1e-6, 1e-5]; // ~7 decimal digits of precision.
687        impl_temp_float![f64: 1e-12, 1e-9, 1e-6]; // ~15–16 decimal digits of precision.
688        #[cfg(feature = "nightly_float")]
689        impl_temp_float![f128: 1e-30, 1e-27, 1e-24]; // ~33–34 decimal digits of precision.
690    };
691    ($f:ty: $lm:literal, $mm:literal, $hm:literal) => {
692        impl TempFloat<$f> {
693            // Practical margins of error.
694            pub const LOW_MARGIN: $f = $lm;
695            pub const MEDIUM_MARGIN: $f = $mm;
696            pub const HIGH_MARGIN: $f = $hm;
697        }
698    };
699}
700impl_temp_float![];
devela/num/float/constants.rs

devela/num/float/
constants.rs
