aanl.lib

//################################ aanl.lib ##########################################
// A library for antialiased nonlinearities. Its official prefix is `aa`. 
//
// This library provides aliasing-suppressed nonlinearities through first-order 
// and second-order approximations of continuous-time signals, functions,
// and convolution based on antiderivatives. This technique is particularly 
// effective if combined with low-factor oversampling, for example, operating
// at 96 kHz or 192 kHz sample-rate.
//
// The library contains trigonometric functions as well as other nonlinear 
// functions such as bounded and unbounded saturators.
//
// Due to their limited domains or ranges, some of these functions may not 
// suitable for audio nonlinear processing or waveshaping, although
// they have been included for completeness. Some other functions,
// for example, tan() and tanh(), are only available with first-order
// antialiasing due to the complexity of the antiderivative of the 
// x * f(x) term, particularly because of the necessity of the dilogarithm 
// function, which requires special implementation.
//
// Future improvements to this library may include an adaptive
// mechanism to set the ill-conditioned cases threshold to improve
// performance in varying cases.
//
// Note that the antialiasing functions introduce a delay in the path,
// respectively half and one-sample delay for first and second-order functions.
//
// Also note that due to division by differences, it is vital to use
// double-precision or more to reduce errors.
//
// The environment identifier for this library is `aa`. After importing
// the standard libraries in Faust, the functions below can be called as `aa.function_name`.
// 
// #### References
//
// * <https://www.dafx.de/paper-archive/2016/dafxpapers/20-DAFx-16_paper_41-PN.pdf>
//########################################################################################

declare name "Antialiased nonlinearities";
declare version "0.1";

ba = library("basics.lib");
ma = library("maths.lib");

//==============================Auxiliary Functions=======================================
//========================================================================================

// ---------- Constant --------------------------------------------------------
MAX = ma.INFINITY;


//-------`(aa.)clip`----------
// Clipping function
//-----------------------------
clip(l, h, x) = max(l, min(h, x));


//-------`(aa.)Rsqrt`----------
// Real-valued sqrt()
//-----------------------------
Rsqrt(x) = sqrt(max(0.0, x));


//-------`(aa.)Rlog`----------
// Real-valued log()
//-----------------------------
Rlog(x) = log(max(ma.EPSILON, x));


//-------`(aa.)Rtan`----------
// Real-valued tan()
//-----------------------------
Rtan(x) = tan(clip(-MAX, MAX, x));


//-------`(aa.)Racos`----------
// Real-valued acos()
//-----------------------------
Racos(x) = acos(clip(-1.0, 1.0, x));


//-------`(aa.)Rasin`----------
// Real-valued asin()
//-----------------------------
Racos(x) = asin(clip(-1.0, 1.0, x));


//-------`(aa.)Racosh`----------
// Real-valued acosh()
//-----------------------------
Racos(x) = ma.acosh(clip(1.0, MAX, x));


//-------`(aa.)Rcosh`----------
// Real-valued cosh()
//-----------------------------
Rcosh(x) = min(MAX, ma.cosh(x));


//-------`(aa.)Rsinh`----------
// Real-valued sinh()
//------------------------------
Rsinh(x) = clip(-MAX, MAX, ma.sinh(x));


//-------`(aa.)Ratanh`----------
// Real-valued atanh()
//------------------------------
Ratanh(x) = ma.atanh(clip(-1.0 + ma.EPSILON, 1.0 - ma.EPSILON, x));


//-------`(aa.)ADAA1`---------------------
//  Generalised first-order ADAA function.
//----------------------------------------

declare ADAA1 author "Dario Sanfilippo";
declare ADAA1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare ADAA1 license "LGPL v3.0 license";
ADAA1(EPS, f, F1, x) = D1q 
      with {
           D1q = ba.if(abs(x - x') <= EPS, ill_D1q, safe_D1q)
               with {
                   ill_D1q = f(x_m);
                   safe_D1q = (F1(x) - F1(x')) / (x - x');
                   x_m = .5 * (x + x');
               };
      };


//-------`(aa.)ADAA2`---------------------
//  Generalised second-order ADAA function.
//----------------------------------------

declare ADAA2 author "Dario Sanfilippo";
declare ADAA2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare ADAA2 license "LGPL v3.0 license";
ADAA2(EPS, f, F1, F2, x) = T1 + T2 
      with {
           T1 = ba.if((x - x') ^ 2.0 <= EPS, ill_T1, safe_T1)
               with {
                   ill_T1 = .5 * f((x + 2.0 * x') / 3.0);
                   safe_T1 = (x * (F1(x) - F1(x')) - (F2(x) - F2(x'))) /
                       ((x - x') ^ 2.0);
               };
           T2 = ba.if((x' - x'') ^ 2.0 <= EPS, ill_T2, safe_T2)
               with {
                   ill_T2 = .5 * f((x'' + 2.0 * x') / 3.0);
                   safe_T2 = (x'' * (F1(x'') - F1(x')) - (F2(x'') - F2(x'))) /
                       ((x'' - x') ^ 2.0);
               };
      };


//==============================Main functions============================================
//========================================================================================

//================= Saturators ===============
//
// These antialiased saturators perform best with high-amplitude input
// signals. If the input is only slightly saturated, hence producing
// negligible aliasing, the trivial saturator may result in a better
// overall output, as noise can be introduced by first and second ADAA
// at low amplitudes. 
//
// Once determining the lowest saturation level for which the antialiased 
// functions perform adequately, it might be sensible to cross-fade
// between the trivial and the antialiased saturators according to the
// amplitude profile of the input signal.
//==============================================

//-------`(aa.)hardclip`---------------------
//
// First-order ADAA hard-clip.
//
// The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
//
// #### Usage
// ```
// _ : aa.hardclip : _;
// ```

declare hardclip author "Dario Sanfilippo";
declare hardclip copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare hardclip license "LGPL v3.0 license";
hardclip(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = clip(-1.0, 1.0, x_f);
           F1(x_F1) = ba.if(   (x_F1 <= 1.0) & (x_F1 >= -1.0), 
                               .5 * x_F1 ^ 2.0, 
                               x_F1 * ma.signum(x_F1) - .5);
      };


//-------`(aa.)hardclip2`---------------------
//
// Second-order ADAA hard-clip.
//
// The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
//
// #### Usage
// ```
// _ : aa.hardclip2 : _;
// ```

declare hardclip2 author "Dario Sanfilippo";
declare hardclip2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare hardclip2 license "LGPL v3.0 license";
hardclip2(x) = ADAA2(EPS, f, F1, F2, x) 
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = clip(-1.0, 1.0, x_f);
           F1(x_F1) = ba.if(   (x_F1 <= 1.0) & (x_F1 >= -1.0), 
                               .5 * x_F1 ^ 2.0, 
                               x_F1 * ma.signum(x_F1) - .5);
           F2(x_F2) = ba.if(   (x_F2 <= 1.0) & (x_F2 >= -1.0), 
                               (1.0 / 3.0) * x_F2 ^ 3.0, 
                               ((.5 * x_F2 ^ 2.0) - 1.0 / 6.0) * 
                                   ma.signum(x_F2));
      };


//-------`(aa.)cubic1`---------------------
//
// First-order ADAA cubic saturator.
//
// The domain of this function is ℝ; its theoretical range is 
// [-2.0/3.0; 2.0/3.0].
//
// #### Usage
// ```
// _ : aa.cubic1 : _;
// ```

declare cubic1 author "Dario Sanfilippo";
declare cubic1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare cubic1 license "LGPL v3.0 license";
cubic1(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = ba.if( (x_f < 1.0) & (x_f > -1.0), 
                           x_f - x_f ^ 3.0 / 3.0, 
                           (2.0 / 3.0) * ma.signum(x_f));
           F1(x_F1) = ba.if(   x_F1 <= -1.0,
                               x_F1 * -2.0 / 3.0,
                               ba.if(  x_F1 >= 1.0,
                                       x_F1 * 2.0 / 3.0,
                                       x_F1 ^ 2.0 / 2.0 - x_F1 ^ 4.0 / 12.0));
      };


//-------`(aa.)parabolic`---------------------
//
// First-order ADAA parabolic saturator.
//
// The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
//
// #### Usage
// ```
// _ : aa.parabolic : _;
// ```

declare parabolic author "Dario Sanfilippo";
declare parabolic copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare parabolic license "LGPL v3.0 license";
parabolic(x) = ADAA1(EPS, f, F1, x) 
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = ba.if( abs(x_f) <= 2.0, 
                           x_f * (1.0 - abs(x_f) / 4.0), 
                           ma.signum(x_f));
           F1(x_F1) = ba.if(   abs(x_F1) <= 2.0, 
                               (-1.0 / 12.0) * x_F1 ^ 2.0 * 
                                   (x_F1 * ma.signum(x_F1) - 6.0),
                               abs(x_F1));
      };


//-------`(aa.)parabolic2`---------------------
//
// Second-order ADAA parabolic saturator.
//
// The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
//
// #### Usage
// ```
// _ : aa.parabolic : _;
// ```

declare parabolic2 author "Dario Sanfilippo";
declare parabolic2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare parabolic2 license "LGPL v3.0 license";
parabolic2(x) = ADAA2(EPS, f, F1, F2, x) 
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = ba.if( abs(x_f) <= 2.0,
                           x_f * (1.0 - abs(x_f) / 4.0),
                           ma.signum(x_f));
           F1(x_F1) = ba.if(   abs(x_F1) <= 2.0,
                               (-1.0 / 12.0) * x_F1 ^ 2.0 * 
                                   (x_F1 * ma.signum(x_F1) - 6.0),
                               abs(x_F1));
           F2(x_F2) = ba.if(   abs(x_F2) <= 2.0,
                               (1.0 / 48.0) * x_F2 ^ 3.0 * 
                                   (16.0 - 3.0 * x_F2 * ma.signum(x_F2)),
                               .5 * x_F2 ^ 2.0 * ma.signum(x_F2));
      };


//-------`(aa.)hyperbolic`---------------------
//
// First-order ADAA hyperbolic saturator.
//
// The domain of this function is ℝ; its theoretical range is ]-1.0; 1.0[.
//
// #### Usage
// ```
// _ : aa.hyperbolic : _;
// ```
declare hyperbolic author "Dario Sanfilippo";
declare hyperbolic copyright "Copyright (C) 2021 Dario Sanfilippo
     <sanfilippo.dario@gmail.com>";
declare hyperbolic license "LGPL v3.0 license";
hyperbolic(x) = ADAA1(EPS, f, F1, x)
    with {
        EPS = 1.0 / ma.SR;
        f(x_f) = x_f / (1.0 + abs(x_f));
        F1(x_F1) = abs(x_F1) - log(1.0 + abs(x_F1));
    };


//-------`(aa.)hyperbolic2`---------------------
//
// Second-order ADAA hyperbolic saturator.
//
// The domain of this function is ℝ; its theoretical range is ]-1.0; 1.0[.
//
// #### Usage
// ```
// _ : aa.hyperbolic2 : _;
// ```
declare hyperbolic2 author "Dario Sanfilippo";
declare hyperbolic2 copyright "Copyright (C) 2021 Dario Sanfilippo
     <sanfilippo.dario@gmail.com>";
declare hyperbolic2 license "LGPL v3.0 license";
hyperbolic2(x) = ADAA2(EPS, f, F1, F2, x)
    with {
        EPS = 1.0 / ma.SR;
        f(x_f) = x_f / (1.0 + abs(x_f));
        F1(x_F1) = abs(x_F1) - log(1.0 + abs(x_F1));
        F2(x_F2) = ma.signum(x_F2) * (.5 * (3.0 - 2.0 * abs(x_F2) + x_F2 ^ 2.0 + 
            2.0 * log(1.0 + abs(x_F2))) - 3.0 / 2.0) + 3.0 / 2.0;
    };


//-------`(aa.)sinarctan`---------------------
//
// First-order ADAA sin(atan()) saturator.
//
// The domain of this function is ℝ; its theoretical range is ]-1.0; 1.0[.
//
// #### Usage
// ```
// _ : aa.sinatan : _;
// ```
declare sinarctan author "Dario Sanfilippo";
declare sinarctan copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare sinarctan license "LGPL v3.0 license";
sinarctan(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = x_f / sqrt(1.0 + x_f ^ 2.0);
           F1(x_F1) = sqrt(x_F1 ^ 2.0 + 1.0) + const
               with {
                   const = -1.0; // for F1(0) = 0 to minimise precision loss
               };
      };


//-------`(aa.)sinarctan2`---------------------
//
// Second-order ADAA sin(atan()) saturator.
//
// The domain of this function is ℝ; its theoretical range is ]-1.0; 1.0[.
//
// #### Usage
// ```
// _ : aa.sinarctan2 : _;
// ```
declare sinarctan2 author "Dario Sanfilippo";
declare sinarctan2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare sinarctan2 license "LGPL v3.0 license";
sinarctan2(x) = ADAA2(EPS, f, F1, F2, x) 
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = x_f / sqrt(1.0 + x_f ^ 2.0);
           F1(x_F1) = sqrt(x_F1 ^ 2.0 + 1.0) + const
               with {
                   const = -1.0; // for F1(0) = 0 to minimise precision loss
               };
           F2(x_F2) = .5 * x_F2 * sqrt(x_F2 ^ 2.0 + 1.0) - .5 * ma.asinh(x_F2);
      };


//-------`(aa.)tanh1`---------------------
//
// First-order ADAA tanh() saturator.
//
// The domain of this function is ℝ; its theoretical range is ]-1.0; 1.0[.
//
// #### Usage
// ```
// _ : aa.tanh1 : _;
// ```
declare tanh1 author "Dario Sanfilippo";
declare tanh1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare tanh1 license "LGPL v3.0 license";
tanh1(x) = ADAA1(EPS, f, F1, x) 
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = ma.tanh(x_f);
           F1(x_F1) = log(Rcosh(x_F1));
      };


//-------`(aa.)arctan`---------------------
//
// First-order ADAA atan().
//
// The domain of this function is ℝ; its theoretical range is ]-π/2.0; π/2.0[.
//
// #### Usage
// ```
// _ : aa.arctan : _;
// ```
declare arctan author "Dario Sanfilippo";
declare arctan copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare arctan license "LGPL v3.0 license";
arctan(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = atan(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - .5 * log(x_F1 ^ 2.0 + 1.0);
      };


//-------`(aa.)arctan2`---------------------
//
// Second-order ADAA atan().
//
// The domain of this function is ℝ; its theoretical range is ]-π/2.0; π/2.0[.
//
// #### Usage
// ```
// _ : aa.arctan2 : _;
// ```
declare arctan2 author "Dario Sanfilippo";
declare arctan2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare arctan2 license "LGPL v3.0 license";
arctan2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = atan(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - .5 * log(x_F1 ^ 2.0 + 1.0);
           F2(x_F2) = .5 * x_F2 ^ 2.0 * f(x_F2) - x_F2 / 2.0 + .5 * f(x_F2);
      };


//-------`(aa.)asinh1`---------------------
//
// First-order ADAA asinh() saturator (unbounded).
//
// The domain of this function is ℝ; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.asinh1 : _;
// ```
declare asinh1 author "Dario Sanfilippo";
declare asinh1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare asinh1 license "LGPL v3.0 license";
asinh1(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = ma.asinh(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - sqrt(1.0 + x_F1 ^ 2.0) + const
               with {
                   const = 1.0; // for F1(0) = 0 to minimise precision loss
               };

      };


//-------`(aa.)asinh2`---------------------
//
// Second-order ADAA asinh() saturator (unbounded).
//
// The domain of this function is ℝ; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.asinh2 : _;
// ```
declare asinh2 author "Dario Sanfilippo";
declare asinh2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare asinh2 license "LGPL v3.0 license";
asinh2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = ma.asinh(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - sqrt(1.0 + x_F1 ^ 2.0) + const
               with {
                   const = 1.0; // for F1(0) = 0 to minimise precision loss
               };
           F2(x_F2) = .25 * (2.0 * x_F2 ^ 2.0 * f(x_F2) + f(x_F2) - 
               sqrt(x_F2 ^ 2.0 + 1.0) * x_F2);

      };


//================= Trigonometry ===============
// These functions are reliable if input signals are within their domains.
//============================================


//-------`(aa.)cosine1`---------------------
//
// First-order ADAA cos().
//
// The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
//
// #### Usage
// ```
// _ : aa.cosine1 : _;
// ```
declare cosine1 author "Dario Sanfilippo";
declare cosine1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare cosine1 license "LGPL v3.0 license";
cosine1(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = cos(x_f);
           F1(x_F1) = sin(x_F1);
      };


//-------`(aa.)cosine2`---------------------
//
// Second-order ADAA cos().
//
// The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
//
// #### Usage
// ```
// _ : aa.cosine2 : _;
// ```
declare cosine2 author "Dario Sanfilippo";
declare cosine2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare cosine2 license "LGPL v3.0 license";
cosine2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = cos(x_f);
           F1(x_F1) = sin(x_F1);
           F2(x_F2) = x_F2 * sin(x_F2) + cos(x_F2);
      };



//-------`(aa.)arccos`---------------------
//
// First-order ADAA acos().
//
// The domain of this function is [-1.0; 1.0]; its theoretical range is
// [π; 0.0].
//
// #### Usage
// ```
// _ : aa.arccos : _;
// ```
declare arccos author "Dario Sanfilippo";
declare arccos copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare arccos license "LGPL v3.0 license";
arccos(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Racos(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - Rsqrt(1.0 - x_F1 ^ 2.0) + const
               with {
                   const = 1.0; // for F1(0) = 0 to minimise precision loss
               };
      };


//-------`(aa.)arccos2`---------------------
//
// Second-order ADAA acos().
//
// The domain of this function is [-1.0; 1.0]; its theoretical range is 
// [π; 0.0].
//
// Note that this function is not accurate for low-amplitude or low-frequency 
// input signals. In that case, the first-order ADAA arccos() can be used.
//
// #### Usage
// ```
// _ : aa.arccos2 : _;
// ```
declare arccos2 author "Dario Sanfilippo";
declare arccos2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare arccos2 license "LGPL v3.0 license";
arccos2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Racos(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - Rsqrt(1.0 - x_F1 ^ 2.0) + const
               with {
                   const = 1.0; // for F1(0) = 0 to minimise precision loss
               };
           F2(x_F2) = .25 * (2.0 * x_F2 ^ 2.0 * f(x_F2) + Rasin(x_F2) -
               Rsqrt(1.0 - x_F2 ^ 2.0));
      };


//-------`(aa.)acosh1`---------------------
//
// First-order ADAA acosh(). 
//
// The domain of this function is ℝ >= 1.0; its theoretical range is ℝ >= 0.0.
//
// #### Usage
// ```
// _ : aa.acosh1 : _;
// ```
declare acosh1 author "Dario Sanfilippo";
declare acosh1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare acosh1 license "LGPL v3.0 license";
acosh1(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Racosh(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - Rsqrt(x_F1 - 1.0) * Rsqrt(x_F1 + 1.0);
      };


//-------`(aa.)acosh2`---------------------
//
// Second-order ADAA acosh().
//
// The domain of this function is ℝ >= 1.0; its theoretical range is ℝ >= 0.0.
//
// Note that this function is not accurate for low-frequency input signals. 
// In that case, the first-order ADAA acosh() can be used.
//
// #### Usage
// ```
// _ : aa.acosh2 : _;
// ```
declare acosh2 author "Dario Sanfilippo";
declare acosh2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare acosh2 license "LGPL v3.0 license";
acosh2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Racosh(x_f);
           F1(x_F1) = x_F1 * f(x_F1) - Rsqrt(x_F1 - 1.0) * Rsqrt(x_F1 + 1.0);
           F2(x_F2) = .5 * x_F2 ^ 2.0 * f(x_F2) - .25 * 
               Rsqrt(x_F2 - 1.0) * Rsqrt(x_F2 + 1.0) * x_F2 -
                   .25 * Rlog(x_F2 + Rsqrt(x_F2 - 1.0) * Rsqrt(x_F2 + 1.0));
      };


//-------`(aa.)sine`---------------------
//
// First-order ADAA sin().
//
// The domain of this function is ℝ; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.sine : _;
// ```
declare sine author "Dario Sanfilippo";
declare sine copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare sine license "LGPL v3.0 license";
sine(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = sin(x_f);
           F1(x_F1) = -1.0 * cos(x_F1);
      };


//-------`(aa.)sine2`---------------------
//
// Second-order ADAA sin().
//
// The domain of this function is ℝ; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.sine2 : _;
// ```
declare sine2 author "Dario Sanfilippo";
declare sine2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare sine2 license "LGPL v3.0 license";
sine2(x) = ADAA2(EPS, f, F1, F2, x) 
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = sin(x_f);
           F1(x_F1) = 1.0 - cos(x_F1);
           F2(x_F2) = sin(x_F2) - x_F2 * cos(x_F2);
      };


//-------`(aa.)arcsin`---------------------
//
// First-order ADAA asin().
//
// The domain of this function is [-1.0, 1.0]; its theoretical range is 
// [-π/2.0; π/2.0].
//
// #### Usage
// ```
// _ : aa.arcsin : _;
// ```
declare arcsin author "Dario Sanfilippo";
declare arcsin copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare arcsin license "LGPL v3.0 license";
arcsin(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Rasin(x_f);
           F1(x_F1) = x_F1 * f(x_F1) + Rsqrt(1.0 - x_F1 ^ 2.0) + const
               with {
                   const = -1.0; // for F1(0) = 0 to minimise precision loss
               };
      };


//-------`(aa.)arcsin2`---------------------
//
// Second-order ADAA asin().
//
// The domain of this function is [-1.0, 1.0]; its theoretical range is
// [-π/2.0; π/2.0].
//
// Note that this function is not accurate for low-frequency input signals.
// In that case, the first-order ADAA asin() can be used.
//
// #### Usage
// ```
// _ : aa.arcsin2 : _;
// ```
declare arcsin2 author "Dario Sanfilippo";
declare arcsin2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare arcsin2 license "LGPL v3.0 license";
arcsin2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Rasin(x_f);
           F1(x_F1) = x_F1 * f(x_F1) + Rsqrt(1.0 - x_F1 ^ 2.0) + const
               with {
                   const = -1.0; // for F1(0) = 0 to minimise precision loss
               };
           F2(x_F2) = .25 * (Rsqrt(1.0 - x_F2 ^ 2.0) * x_F2 + 
               (2.0 * x_F2 ^ 2.0 - 1.0) * f(x_F2));
      };


//-------`(aa.)tangent`---------------------
//
// First-order ADAA tan().
//
// The domain of this function is [-π/2.0; π/2.0]; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.tangent : _;
// ```
declare tangent author "Dario Sanfilippo";
declare tangent copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare tangent license "LGPL v3.0 license";
tangent(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Rtan(x_f);
           F1(x_F1) = -1.0 * Rlog(cos(x_F1));
      };


//-------`(aa.)atanh1`---------------------
//
// First-order ADAA atanh(). 
//
// The domain of this function is ]-1.0; 1.0[; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.atanh1 : _;
// ```
declare atanh1 author "Dario Sanfilippo";
declare atanh1 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare atanh1 license "LGPL v3.0 license";
atanh1(x) = ADAA1(EPS, f, F1, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Ratanh(x_f);
           F1(x_F1) = .5 * Rlog(1.0 - x_F1 ^ 2.0) + x_F1 * f(x_F1);
      };


//-------`(aa.)atanh2`---------------------
//
// Second-order ADAA atanh().
//
// The domain of this function is ]-1.0; 1.0[; its theoretical range is ℝ.
//
// #### Usage
// ```
// _ : aa.atanh2 : _;
// ```
declare atanh2 author "Dario Sanfilippo";
declare atanh2 copyright "Copyright (C) 2021 Dario Sanfilippo
    <sanfilippo.dario@gmail.com>";
declare atanh2 license "LGPL v3.0 license";
atanh2(x) = ADAA2(EPS, f, F1, F2, x)
      with {
           EPS = 1.0 / ma.SR;
           f(x_f) = Ratanh(x_f);
           F1(x_F1) = .5 * Rlog(1.0 - x_F1 ^ 2.0) + x_F1 * f(x_F1);
           F2(x_F2) = .5 * x_F2 ^ 2.0 * f(x_F2) + x_F2 / 2.0 + .25 * 
               Rlog(1.0 - x_F2) - .25 * Rlog(1.0 + x_F2);
      };