1 #![cfg(any(feature = "std", feature = "libm"))]
2 
3 use core::ops::Neg;
4 
5 use {Float, Num, NumCast};
6 
7 // NOTE: These doctests have the same issue as those in src/float.rs.
8 // They're testing the inherent methods directly, and not those of `Real`.
9 
10 /// A trait for real number types that do not necessarily have
11 /// floating-point-specific characteristics such as NaN and infinity.
12 ///
13 /// See [this Wikipedia article](https://en.wikipedia.org/wiki/Real_data_type)
14 /// for a list of data types that could meaningfully implement this trait.
15 ///
16 /// This trait is only available with the `std` feature, or with the `libm` feature otherwise.
17 pub trait Real: Num + Copy + NumCast + PartialOrd + Neg<Output = Self> {
18     /// Returns the smallest finite value that this type can represent.
19     ///
20     /// ```
21     /// use num_traits::real::Real;
22     /// use std::f64;
23     ///
24     /// let x: f64 = Real::min_value();
25     ///
26     /// assert_eq!(x, f64::MIN);
27     /// ```
min_value() -> Self28     fn min_value() -> Self;
29 
30     /// Returns the smallest positive, normalized value that this type can represent.
31     ///
32     /// ```
33     /// use num_traits::real::Real;
34     /// use std::f64;
35     ///
36     /// let x: f64 = Real::min_positive_value();
37     ///
38     /// assert_eq!(x, f64::MIN_POSITIVE);
39     /// ```
min_positive_value() -> Self40     fn min_positive_value() -> Self;
41 
42     /// Returns epsilon, a small positive value.
43     ///
44     /// ```
45     /// use num_traits::real::Real;
46     /// use std::f64;
47     ///
48     /// let x: f64 = Real::epsilon();
49     ///
50     /// assert_eq!(x, f64::EPSILON);
51     /// ```
52     ///
53     /// # Panics
54     ///
55     /// The default implementation will panic if `f32::EPSILON` cannot
56     /// be cast to `Self`.
epsilon() -> Self57     fn epsilon() -> Self;
58 
59     /// Returns the largest finite value that this type can represent.
60     ///
61     /// ```
62     /// use num_traits::real::Real;
63     /// use std::f64;
64     ///
65     /// let x: f64 = Real::max_value();
66     /// assert_eq!(x, f64::MAX);
67     /// ```
max_value() -> Self68     fn max_value() -> Self;
69 
70     /// Returns the largest integer less than or equal to a number.
71     ///
72     /// ```
73     /// use num_traits::real::Real;
74     ///
75     /// let f = 3.99;
76     /// let g = 3.0;
77     ///
78     /// assert_eq!(f.floor(), 3.0);
79     /// assert_eq!(g.floor(), 3.0);
80     /// ```
floor(self) -> Self81     fn floor(self) -> Self;
82 
83     /// Returns the smallest integer greater than or equal to a number.
84     ///
85     /// ```
86     /// use num_traits::real::Real;
87     ///
88     /// let f = 3.01;
89     /// let g = 4.0;
90     ///
91     /// assert_eq!(f.ceil(), 4.0);
92     /// assert_eq!(g.ceil(), 4.0);
93     /// ```
ceil(self) -> Self94     fn ceil(self) -> Self;
95 
96     /// Returns the nearest integer to a number. Round half-way cases away from
97     /// `0.0`.
98     ///
99     /// ```
100     /// use num_traits::real::Real;
101     ///
102     /// let f = 3.3;
103     /// let g = -3.3;
104     ///
105     /// assert_eq!(f.round(), 3.0);
106     /// assert_eq!(g.round(), -3.0);
107     /// ```
round(self) -> Self108     fn round(self) -> Self;
109 
110     /// Return the integer part of a number.
111     ///
112     /// ```
113     /// use num_traits::real::Real;
114     ///
115     /// let f = 3.3;
116     /// let g = -3.7;
117     ///
118     /// assert_eq!(f.trunc(), 3.0);
119     /// assert_eq!(g.trunc(), -3.0);
120     /// ```
trunc(self) -> Self121     fn trunc(self) -> Self;
122 
123     /// Returns the fractional part of a number.
124     ///
125     /// ```
126     /// use num_traits::real::Real;
127     ///
128     /// let x = 3.5;
129     /// let y = -3.5;
130     /// let abs_difference_x = (x.fract() - 0.5).abs();
131     /// let abs_difference_y = (y.fract() - (-0.5)).abs();
132     ///
133     /// assert!(abs_difference_x < 1e-10);
134     /// assert!(abs_difference_y < 1e-10);
135     /// ```
fract(self) -> Self136     fn fract(self) -> Self;
137 
138     /// Computes the absolute value of `self`. Returns `Float::nan()` if the
139     /// number is `Float::nan()`.
140     ///
141     /// ```
142     /// use num_traits::real::Real;
143     /// use std::f64;
144     ///
145     /// let x = 3.5;
146     /// let y = -3.5;
147     ///
148     /// let abs_difference_x = (x.abs() - x).abs();
149     /// let abs_difference_y = (y.abs() - (-y)).abs();
150     ///
151     /// assert!(abs_difference_x < 1e-10);
152     /// assert!(abs_difference_y < 1e-10);
153     ///
154     /// assert!(::num_traits::Float::is_nan(f64::NAN.abs()));
155     /// ```
abs(self) -> Self156     fn abs(self) -> Self;
157 
158     /// Returns a number that represents the sign of `self`.
159     ///
160     /// - `1.0` if the number is positive, `+0.0` or `Float::infinity()`
161     /// - `-1.0` if the number is negative, `-0.0` or `Float::neg_infinity()`
162     /// - `Float::nan()` if the number is `Float::nan()`
163     ///
164     /// ```
165     /// use num_traits::real::Real;
166     /// use std::f64;
167     ///
168     /// let f = 3.5;
169     ///
170     /// assert_eq!(f.signum(), 1.0);
171     /// assert_eq!(f64::NEG_INFINITY.signum(), -1.0);
172     ///
173     /// assert!(f64::NAN.signum().is_nan());
174     /// ```
signum(self) -> Self175     fn signum(self) -> Self;
176 
177     /// Returns `true` if `self` is positive, including `+0.0`,
178     /// `Float::infinity()`, and with newer versions of Rust `f64::NAN`.
179     ///
180     /// ```
181     /// use num_traits::real::Real;
182     /// use std::f64;
183     ///
184     /// let neg_nan: f64 = -f64::NAN;
185     ///
186     /// let f = 7.0;
187     /// let g = -7.0;
188     ///
189     /// assert!(f.is_sign_positive());
190     /// assert!(!g.is_sign_positive());
191     /// assert!(!neg_nan.is_sign_positive());
192     /// ```
is_sign_positive(self) -> bool193     fn is_sign_positive(self) -> bool;
194 
195     /// Returns `true` if `self` is negative, including `-0.0`,
196     /// `Float::neg_infinity()`, and with newer versions of Rust `-f64::NAN`.
197     ///
198     /// ```
199     /// use num_traits::real::Real;
200     /// use std::f64;
201     ///
202     /// let nan: f64 = f64::NAN;
203     ///
204     /// let f = 7.0;
205     /// let g = -7.0;
206     ///
207     /// assert!(!f.is_sign_negative());
208     /// assert!(g.is_sign_negative());
209     /// assert!(!nan.is_sign_negative());
210     /// ```
is_sign_negative(self) -> bool211     fn is_sign_negative(self) -> bool;
212 
213     /// Fused multiply-add. Computes `(self * a) + b` with only one rounding
214     /// error, yielding a more accurate result than an unfused multiply-add.
215     ///
216     /// Using `mul_add` can be more performant than an unfused multiply-add if
217     /// the target architecture has a dedicated `fma` CPU instruction.
218     ///
219     /// ```
220     /// use num_traits::real::Real;
221     ///
222     /// let m = 10.0;
223     /// let x = 4.0;
224     /// let b = 60.0;
225     ///
226     /// // 100.0
227     /// let abs_difference = (m.mul_add(x, b) - (m*x + b)).abs();
228     ///
229     /// assert!(abs_difference < 1e-10);
230     /// ```
mul_add(self, a: Self, b: Self) -> Self231     fn mul_add(self, a: Self, b: Self) -> Self;
232 
233     /// Take the reciprocal (inverse) of a number, `1/x`.
234     ///
235     /// ```
236     /// use num_traits::real::Real;
237     ///
238     /// let x = 2.0;
239     /// let abs_difference = (x.recip() - (1.0/x)).abs();
240     ///
241     /// assert!(abs_difference < 1e-10);
242     /// ```
recip(self) -> Self243     fn recip(self) -> Self;
244 
245     /// Raise a number to an integer power.
246     ///
247     /// Using this function is generally faster than using `powf`
248     ///
249     /// ```
250     /// use num_traits::real::Real;
251     ///
252     /// let x = 2.0;
253     /// let abs_difference = (x.powi(2) - x*x).abs();
254     ///
255     /// assert!(abs_difference < 1e-10);
256     /// ```
powi(self, n: i32) -> Self257     fn powi(self, n: i32) -> Self;
258 
259     /// Raise a number to a real number power.
260     ///
261     /// ```
262     /// use num_traits::real::Real;
263     ///
264     /// let x = 2.0;
265     /// let abs_difference = (x.powf(2.0) - x*x).abs();
266     ///
267     /// assert!(abs_difference < 1e-10);
268     /// ```
powf(self, n: Self) -> Self269     fn powf(self, n: Self) -> Self;
270 
271     /// Take the square root of a number.
272     ///
273     /// Returns NaN if `self` is a negative floating-point number.
274     ///
275     /// # Panics
276     ///
277     /// If the implementing type doesn't support NaN, this method should panic if `self < 0`.
278     ///
279     /// ```
280     /// use num_traits::real::Real;
281     ///
282     /// let positive = 4.0;
283     /// let negative = -4.0;
284     ///
285     /// let abs_difference = (positive.sqrt() - 2.0).abs();
286     ///
287     /// assert!(abs_difference < 1e-10);
288     /// assert!(::num_traits::Float::is_nan(negative.sqrt()));
289     /// ```
sqrt(self) -> Self290     fn sqrt(self) -> Self;
291 
292     /// Returns `e^(self)`, (the exponential function).
293     ///
294     /// ```
295     /// use num_traits::real::Real;
296     ///
297     /// let one = 1.0;
298     /// // e^1
299     /// let e = one.exp();
300     ///
301     /// // ln(e) - 1 == 0
302     /// let abs_difference = (e.ln() - 1.0).abs();
303     ///
304     /// assert!(abs_difference < 1e-10);
305     /// ```
exp(self) -> Self306     fn exp(self) -> Self;
307 
308     /// Returns `2^(self)`.
309     ///
310     /// ```
311     /// use num_traits::real::Real;
312     ///
313     /// let f = 2.0;
314     ///
315     /// // 2^2 - 4 == 0
316     /// let abs_difference = (f.exp2() - 4.0).abs();
317     ///
318     /// assert!(abs_difference < 1e-10);
319     /// ```
exp2(self) -> Self320     fn exp2(self) -> Self;
321 
322     /// Returns the natural logarithm of the number.
323     ///
324     /// # Panics
325     ///
326     /// If `self <= 0` and this type does not support a NaN representation, this function should panic.
327     ///
328     /// ```
329     /// use num_traits::real::Real;
330     ///
331     /// let one = 1.0;
332     /// // e^1
333     /// let e = one.exp();
334     ///
335     /// // ln(e) - 1 == 0
336     /// let abs_difference = (e.ln() - 1.0).abs();
337     ///
338     /// assert!(abs_difference < 1e-10);
339     /// ```
ln(self) -> Self340     fn ln(self) -> Self;
341 
342     /// Returns the logarithm of the number with respect to an arbitrary base.
343     ///
344     /// # Panics
345     ///
346     /// If `self <= 0` and this type does not support a NaN representation, this function should panic.
347     ///
348     /// ```
349     /// use num_traits::real::Real;
350     ///
351     /// let ten = 10.0;
352     /// let two = 2.0;
353     ///
354     /// // log10(10) - 1 == 0
355     /// let abs_difference_10 = (ten.log(10.0) - 1.0).abs();
356     ///
357     /// // log2(2) - 1 == 0
358     /// let abs_difference_2 = (two.log(2.0) - 1.0).abs();
359     ///
360     /// assert!(abs_difference_10 < 1e-10);
361     /// assert!(abs_difference_2 < 1e-10);
362     /// ```
log(self, base: Self) -> Self363     fn log(self, base: Self) -> Self;
364 
365     /// Returns the base 2 logarithm of the number.
366     ///
367     /// # Panics
368     ///
369     /// If `self <= 0` and this type does not support a NaN representation, this function should panic.
370     ///
371     /// ```
372     /// use num_traits::real::Real;
373     ///
374     /// let two = 2.0;
375     ///
376     /// // log2(2) - 1 == 0
377     /// let abs_difference = (two.log2() - 1.0).abs();
378     ///
379     /// assert!(abs_difference < 1e-10);
380     /// ```
log2(self) -> Self381     fn log2(self) -> Self;
382 
383     /// Returns the base 10 logarithm of the number.
384     ///
385     /// # Panics
386     ///
387     /// If `self <= 0` and this type does not support a NaN representation, this function should panic.
388     ///
389     ///
390     /// ```
391     /// use num_traits::real::Real;
392     ///
393     /// let ten = 10.0;
394     ///
395     /// // log10(10) - 1 == 0
396     /// let abs_difference = (ten.log10() - 1.0).abs();
397     ///
398     /// assert!(abs_difference < 1e-10);
399     /// ```
log10(self) -> Self400     fn log10(self) -> Self;
401 
402     /// Converts radians to degrees.
403     ///
404     /// ```
405     /// use std::f64::consts;
406     ///
407     /// let angle = consts::PI;
408     ///
409     /// let abs_difference = (angle.to_degrees() - 180.0).abs();
410     ///
411     /// assert!(abs_difference < 1e-10);
412     /// ```
to_degrees(self) -> Self413     fn to_degrees(self) -> Self;
414 
415     /// Converts degrees to radians.
416     ///
417     /// ```
418     /// use std::f64::consts;
419     ///
420     /// let angle = 180.0_f64;
421     ///
422     /// let abs_difference = (angle.to_radians() - consts::PI).abs();
423     ///
424     /// assert!(abs_difference < 1e-10);
425     /// ```
to_radians(self) -> Self426     fn to_radians(self) -> Self;
427 
428     /// Returns the maximum of the two numbers.
429     ///
430     /// ```
431     /// use num_traits::real::Real;
432     ///
433     /// let x = 1.0;
434     /// let y = 2.0;
435     ///
436     /// assert_eq!(x.max(y), y);
437     /// ```
max(self, other: Self) -> Self438     fn max(self, other: Self) -> Self;
439 
440     /// Returns the minimum of the two numbers.
441     ///
442     /// ```
443     /// use num_traits::real::Real;
444     ///
445     /// let x = 1.0;
446     /// let y = 2.0;
447     ///
448     /// assert_eq!(x.min(y), x);
449     /// ```
min(self, other: Self) -> Self450     fn min(self, other: Self) -> Self;
451 
452     /// The positive difference of two numbers.
453     ///
454     /// * If `self <= other`: `0:0`
455     /// * Else: `self - other`
456     ///
457     /// ```
458     /// use num_traits::real::Real;
459     ///
460     /// let x = 3.0;
461     /// let y = -3.0;
462     ///
463     /// let abs_difference_x = (x.abs_sub(1.0) - 2.0).abs();
464     /// let abs_difference_y = (y.abs_sub(1.0) - 0.0).abs();
465     ///
466     /// assert!(abs_difference_x < 1e-10);
467     /// assert!(abs_difference_y < 1e-10);
468     /// ```
abs_sub(self, other: Self) -> Self469     fn abs_sub(self, other: Self) -> Self;
470 
471     /// Take the cubic root of a number.
472     ///
473     /// ```
474     /// use num_traits::real::Real;
475     ///
476     /// let x = 8.0;
477     ///
478     /// // x^(1/3) - 2 == 0
479     /// let abs_difference = (x.cbrt() - 2.0).abs();
480     ///
481     /// assert!(abs_difference < 1e-10);
482     /// ```
cbrt(self) -> Self483     fn cbrt(self) -> Self;
484 
485     /// Calculate the length of the hypotenuse of a right-angle triangle given
486     /// legs of length `x` and `y`.
487     ///
488     /// ```
489     /// use num_traits::real::Real;
490     ///
491     /// let x = 2.0;
492     /// let y = 3.0;
493     ///
494     /// // sqrt(x^2 + y^2)
495     /// let abs_difference = (x.hypot(y) - (x.powi(2) + y.powi(2)).sqrt()).abs();
496     ///
497     /// assert!(abs_difference < 1e-10);
498     /// ```
hypot(self, other: Self) -> Self499     fn hypot(self, other: Self) -> Self;
500 
501     /// Computes the sine of a number (in radians).
502     ///
503     /// ```
504     /// use num_traits::real::Real;
505     /// use std::f64;
506     ///
507     /// let x = f64::consts::PI/2.0;
508     ///
509     /// let abs_difference = (x.sin() - 1.0).abs();
510     ///
511     /// assert!(abs_difference < 1e-10);
512     /// ```
sin(self) -> Self513     fn sin(self) -> Self;
514 
515     /// Computes the cosine of a number (in radians).
516     ///
517     /// ```
518     /// use num_traits::real::Real;
519     /// use std::f64;
520     ///
521     /// let x = 2.0*f64::consts::PI;
522     ///
523     /// let abs_difference = (x.cos() - 1.0).abs();
524     ///
525     /// assert!(abs_difference < 1e-10);
526     /// ```
cos(self) -> Self527     fn cos(self) -> Self;
528 
529     /// Computes the tangent of a number (in radians).
530     ///
531     /// ```
532     /// use num_traits::real::Real;
533     /// use std::f64;
534     ///
535     /// let x = f64::consts::PI/4.0;
536     /// let abs_difference = (x.tan() - 1.0).abs();
537     ///
538     /// assert!(abs_difference < 1e-14);
539     /// ```
tan(self) -> Self540     fn tan(self) -> Self;
541 
542     /// Computes the arcsine of a number. Return value is in radians in
543     /// the range [-pi/2, pi/2] or NaN if the number is outside the range
544     /// [-1, 1].
545     ///
546     /// # Panics
547     ///
548     /// If this type does not support a NaN representation, this function should panic
549     /// if the number is outside the range [-1, 1].
550     ///
551     /// ```
552     /// use num_traits::real::Real;
553     /// use std::f64;
554     ///
555     /// let f = f64::consts::PI / 2.0;
556     ///
557     /// // asin(sin(pi/2))
558     /// let abs_difference = (f.sin().asin() - f64::consts::PI / 2.0).abs();
559     ///
560     /// assert!(abs_difference < 1e-10);
561     /// ```
asin(self) -> Self562     fn asin(self) -> Self;
563 
564     /// Computes the arccosine of a number. Return value is in radians in
565     /// the range [0, pi] or NaN if the number is outside the range
566     /// [-1, 1].
567     ///
568     /// # Panics
569     ///
570     /// If this type does not support a NaN representation, this function should panic
571     /// if the number is outside the range [-1, 1].
572     ///
573     /// ```
574     /// use num_traits::real::Real;
575     /// use std::f64;
576     ///
577     /// let f = f64::consts::PI / 4.0;
578     ///
579     /// // acos(cos(pi/4))
580     /// let abs_difference = (f.cos().acos() - f64::consts::PI / 4.0).abs();
581     ///
582     /// assert!(abs_difference < 1e-10);
583     /// ```
acos(self) -> Self584     fn acos(self) -> Self;
585 
586     /// Computes the arctangent of a number. Return value is in radians in the
587     /// range [-pi/2, pi/2];
588     ///
589     /// ```
590     /// use num_traits::real::Real;
591     ///
592     /// let f = 1.0;
593     ///
594     /// // atan(tan(1))
595     /// let abs_difference = (f.tan().atan() - 1.0).abs();
596     ///
597     /// assert!(abs_difference < 1e-10);
598     /// ```
atan(self) -> Self599     fn atan(self) -> Self;
600 
601     /// Computes the four quadrant arctangent of `self` (`y`) and `other` (`x`).
602     ///
603     /// * `x = 0`, `y = 0`: `0`
604     /// * `x >= 0`: `arctan(y/x)` -> `[-pi/2, pi/2]`
605     /// * `y >= 0`: `arctan(y/x) + pi` -> `(pi/2, pi]`
606     /// * `y < 0`: `arctan(y/x) - pi` -> `(-pi, -pi/2)`
607     ///
608     /// ```
609     /// use num_traits::real::Real;
610     /// use std::f64;
611     ///
612     /// let pi = f64::consts::PI;
613     /// // All angles from horizontal right (+x)
614     /// // 45 deg counter-clockwise
615     /// let x1 = 3.0;
616     /// let y1 = -3.0;
617     ///
618     /// // 135 deg clockwise
619     /// let x2 = -3.0;
620     /// let y2 = 3.0;
621     ///
622     /// let abs_difference_1 = (y1.atan2(x1) - (-pi/4.0)).abs();
623     /// let abs_difference_2 = (y2.atan2(x2) - 3.0*pi/4.0).abs();
624     ///
625     /// assert!(abs_difference_1 < 1e-10);
626     /// assert!(abs_difference_2 < 1e-10);
627     /// ```
atan2(self, other: Self) -> Self628     fn atan2(self, other: Self) -> Self;
629 
630     /// Simultaneously computes the sine and cosine of the number, `x`. Returns
631     /// `(sin(x), cos(x))`.
632     ///
633     /// ```
634     /// use num_traits::real::Real;
635     /// use std::f64;
636     ///
637     /// let x = f64::consts::PI/4.0;
638     /// let f = x.sin_cos();
639     ///
640     /// let abs_difference_0 = (f.0 - x.sin()).abs();
641     /// let abs_difference_1 = (f.1 - x.cos()).abs();
642     ///
643     /// assert!(abs_difference_0 < 1e-10);
644     /// assert!(abs_difference_0 < 1e-10);
645     /// ```
sin_cos(self) -> (Self, Self)646     fn sin_cos(self) -> (Self, Self);
647 
648     /// Returns `e^(self) - 1` in a way that is accurate even if the
649     /// number is close to zero.
650     ///
651     /// ```
652     /// use num_traits::real::Real;
653     ///
654     /// let x = 7.0;
655     ///
656     /// // e^(ln(7)) - 1
657     /// let abs_difference = (x.ln().exp_m1() - 6.0).abs();
658     ///
659     /// assert!(abs_difference < 1e-10);
660     /// ```
exp_m1(self) -> Self661     fn exp_m1(self) -> Self;
662 
663     /// Returns `ln(1+n)` (natural logarithm) more accurately than if
664     /// the operations were performed separately.
665     ///
666     /// # Panics
667     ///
668     /// If this type does not support a NaN representation, this function should panic
669     /// if `self-1 <= 0`.
670     ///
671     /// ```
672     /// use num_traits::real::Real;
673     /// use std::f64;
674     ///
675     /// let x = f64::consts::E - 1.0;
676     ///
677     /// // ln(1 + (e - 1)) == ln(e) == 1
678     /// let abs_difference = (x.ln_1p() - 1.0).abs();
679     ///
680     /// assert!(abs_difference < 1e-10);
681     /// ```
ln_1p(self) -> Self682     fn ln_1p(self) -> Self;
683 
684     /// Hyperbolic sine function.
685     ///
686     /// ```
687     /// use num_traits::real::Real;
688     /// use std::f64;
689     ///
690     /// let e = f64::consts::E;
691     /// let x = 1.0;
692     ///
693     /// let f = x.sinh();
694     /// // Solving sinh() at 1 gives `(e^2-1)/(2e)`
695     /// let g = (e*e - 1.0)/(2.0*e);
696     /// let abs_difference = (f - g).abs();
697     ///
698     /// assert!(abs_difference < 1e-10);
699     /// ```
sinh(self) -> Self700     fn sinh(self) -> Self;
701 
702     /// Hyperbolic cosine function.
703     ///
704     /// ```
705     /// use num_traits::real::Real;
706     /// use std::f64;
707     ///
708     /// let e = f64::consts::E;
709     /// let x = 1.0;
710     /// let f = x.cosh();
711     /// // Solving cosh() at 1 gives this result
712     /// let g = (e*e + 1.0)/(2.0*e);
713     /// let abs_difference = (f - g).abs();
714     ///
715     /// // Same result
716     /// assert!(abs_difference < 1.0e-10);
717     /// ```
cosh(self) -> Self718     fn cosh(self) -> Self;
719 
720     /// Hyperbolic tangent function.
721     ///
722     /// ```
723     /// use num_traits::real::Real;
724     /// use std::f64;
725     ///
726     /// let e = f64::consts::E;
727     /// let x = 1.0;
728     ///
729     /// let f = x.tanh();
730     /// // Solving tanh() at 1 gives `(1 - e^(-2))/(1 + e^(-2))`
731     /// let g = (1.0 - e.powi(-2))/(1.0 + e.powi(-2));
732     /// let abs_difference = (f - g).abs();
733     ///
734     /// assert!(abs_difference < 1.0e-10);
735     /// ```
tanh(self) -> Self736     fn tanh(self) -> Self;
737 
738     /// Inverse hyperbolic sine function.
739     ///
740     /// ```
741     /// use num_traits::real::Real;
742     ///
743     /// let x = 1.0;
744     /// let f = x.sinh().asinh();
745     ///
746     /// let abs_difference = (f - x).abs();
747     ///
748     /// assert!(abs_difference < 1.0e-10);
749     /// ```
asinh(self) -> Self750     fn asinh(self) -> Self;
751 
752     /// Inverse hyperbolic cosine function.
753     ///
754     /// ```
755     /// use num_traits::real::Real;
756     ///
757     /// let x = 1.0;
758     /// let f = x.cosh().acosh();
759     ///
760     /// let abs_difference = (f - x).abs();
761     ///
762     /// assert!(abs_difference < 1.0e-10);
763     /// ```
acosh(self) -> Self764     fn acosh(self) -> Self;
765 
766     /// Inverse hyperbolic tangent function.
767     ///
768     /// ```
769     /// use num_traits::real::Real;
770     /// use std::f64;
771     ///
772     /// let e = f64::consts::E;
773     /// let f = e.tanh().atanh();
774     ///
775     /// let abs_difference = (f - e).abs();
776     ///
777     /// assert!(abs_difference < 1.0e-10);
778     /// ```
atanh(self) -> Self779     fn atanh(self) -> Self;
780 }
781 
782 impl<T: Float> Real for T {
783     forward! {
784         Float::min_value() -> Self;
785         Float::min_positive_value() -> Self;
786         Float::epsilon() -> Self;
787         Float::max_value() -> Self;
788     }
789     forward! {
790         Float::floor(self) -> Self;
791         Float::ceil(self) -> Self;
792         Float::round(self) -> Self;
793         Float::trunc(self) -> Self;
794         Float::fract(self) -> Self;
795         Float::abs(self) -> Self;
796         Float::signum(self) -> Self;
797         Float::is_sign_positive(self) -> bool;
798         Float::is_sign_negative(self) -> bool;
799         Float::mul_add(self, a: Self, b: Self) -> Self;
800         Float::recip(self) -> Self;
801         Float::powi(self, n: i32) -> Self;
802         Float::powf(self, n: Self) -> Self;
803         Float::sqrt(self) -> Self;
804         Float::exp(self) -> Self;
805         Float::exp2(self) -> Self;
806         Float::ln(self) -> Self;
807         Float::log(self, base: Self) -> Self;
808         Float::log2(self) -> Self;
809         Float::log10(self) -> Self;
810         Float::to_degrees(self) -> Self;
811         Float::to_radians(self) -> Self;
812         Float::max(self, other: Self) -> Self;
813         Float::min(self, other: Self) -> Self;
814         Float::abs_sub(self, other: Self) -> Self;
815         Float::cbrt(self) -> Self;
816         Float::hypot(self, other: Self) -> Self;
817         Float::sin(self) -> Self;
818         Float::cos(self) -> Self;
819         Float::tan(self) -> Self;
820         Float::asin(self) -> Self;
821         Float::acos(self) -> Self;
822         Float::atan(self) -> Self;
823         Float::atan2(self, other: Self) -> Self;
824         Float::sin_cos(self) -> (Self, Self);
825         Float::exp_m1(self) -> Self;
826         Float::ln_1p(self) -> Self;
827         Float::sinh(self) -> Self;
828         Float::cosh(self) -> Self;
829         Float::tanh(self) -> Self;
830         Float::asinh(self) -> Self;
831         Float::acosh(self) -> Self;
832         Float::atanh(self) -> Self;
833     }
834 }
835