// Copyright 2010 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package math

/*
	Floating-point logarithm of the Gamma function.
*/

// The original C code and the long comment below are
// from FreeBSD's /usr/src/lib/msun/src/e_lgamma_r.c and
// came with this notice. The go code is a simplified
// version of the original C.
//
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunPro, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// __ieee754_lgamma_r(x, signgamp)
// Reentrant version of the logarithm of the Gamma function
// with user provided pointer for the sign of Gamma(x).
//
// Method:
//   1. Argument Reduction for 0 < x <= 8
//      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
//      reduce x to a number in [1.5,2.5] by
//              lgamma(1+s) = log(s) + lgamma(s)
//      for example,
//              lgamma(7.3) = log(6.3) + lgamma(6.3)
//                          = log(6.3*5.3) + lgamma(5.3)
//                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
//   2. Polynomial approximation of lgamma around its
//      minimum (ymin=1.461632144968362245) to maintain monotonicity.
//      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
//              Let z = x-ymin;
//              lgamma(x) = -1.214862905358496078218 + z**2*poly(z)
//              poly(z) is a 14 degree polynomial.
//   2. Rational approximation in the primary interval [2,3]
//      We use the following approximation:
//              s = x-2.0;
//              lgamma(x) = 0.5*s + s*P(s)/Q(s)
//      with accuracy
//              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
//      Our algorithms are based on the following observation
//
//                             zeta(2)-1    2    zeta(3)-1    3
// lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
//                                 2                 3
//
//      where Euler = 0.5772156649... is the Euler constant, which
//      is very close to 0.5.
//
//   3. For x>=8, we have
//      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
//      (better formula:
//         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
//      Let z = 1/x, then we approximation
//              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
//      by
//                                  3       5             11
//              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
//      where
//              |w - f(z)| < 2**-58.74
//
//   4. For negative x, since (G is gamma function)
//              -x*G(-x)*G(x) = pi/sin(pi*x),
//      we have
//              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
//      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
//      Hence, for x<0, signgam = sign(sin(pi*x)) and
//              lgamma(x) = log(|Gamma(x)|)
//                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
//      Note: one should avoid computing pi*(-x) directly in the
//            computation of sin(pi*(-x)).
//
//   5. Special Cases
//              lgamma(2+s) ~ s*(1-Euler) for tiny s
//              lgamma(1)=lgamma(2)=0
//              lgamma(x) ~ -log(x) for tiny x
//              lgamma(0) = lgamma(inf) = inf
//              lgamma(-integer) = +-inf
//
//

var _lgamA = [...]float64{
	7.72156649015328655494e-02, // 0x3FB3C467E37DB0C8
	3.22467033424113591611e-01, // 0x3FD4A34CC4A60FAD
	6.73523010531292681824e-02, // 0x3FB13E001A5562A7
	2.05808084325167332806e-02, // 0x3F951322AC92547B
	7.38555086081402883957e-03, // 0x3F7E404FB68FEFE8
	2.89051383673415629091e-03, // 0x3F67ADD8CCB7926B
	1.19270763183362067845e-03, // 0x3F538A94116F3F5D
	5.10069792153511336608e-04, // 0x3F40B6C689B99C00
	2.20862790713908385557e-04, // 0x3F2CF2ECED10E54D
	1.08011567247583939954e-04, // 0x3F1C5088987DFB07
	2.52144565451257326939e-05, // 0x3EFA7074428CFA52
	4.48640949618915160150e-05, // 0x3F07858E90A45837
}
var _lgamR = [...]float64{
	1.0,                        // placeholder
	1.39200533467621045958e+00, // 0x3FF645A762C4AB74
	7.21935547567138069525e-01, // 0x3FE71A1893D3DCDC
	1.71933865632803078993e-01, // 0x3FC601EDCCFBDF27
	1.86459191715652901344e-02, // 0x3F9317EA742ED475
	7.77942496381893596434e-04, // 0x3F497DDACA41A95B
	7.32668430744625636189e-06, // 0x3EDEBAF7A5B38140
}
var _lgamS = [...]float64{
	-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
	2.14982415960608852501e-01,  // 0x3FCB848B36E20878
	3.25778796408930981787e-01,  // 0x3FD4D98F4F139F59
	1.46350472652464452805e-01,  // 0x3FC2BB9CBEE5F2F7
	2.66422703033638609560e-02,  // 0x3F9B481C7E939961
	1.84028451407337715652e-03,  // 0x3F5E26B67368F239
	3.19475326584100867617e-05,  // 0x3F00BFECDD17E945
}
var _lgamT = [...]float64{
	4.83836122723810047042e-01,  // 0x3FDEF72BC8EE38A2
	-1.47587722994593911752e-01, // 0xBFC2E4278DC6C509
	6.46249402391333854778e-02,  // 0x3FB08B4294D5419B
	-3.27885410759859649565e-02, // 0xBFA0C9A8DF35B713
	1.79706750811820387126e-02,  // 0x3F9266E7970AF9EC
	-1.03142241298341437450e-02, // 0xBF851F9FBA91EC6A
	6.10053870246291332635e-03,  // 0x3F78FCE0E370E344
	-3.68452016781138256760e-03, // 0xBF6E2EFFB3E914D7
	2.25964780900612472250e-03,  // 0x3F6282D32E15C915
	-1.40346469989232843813e-03, // 0xBF56FE8EBF2D1AF1
	8.81081882437654011382e-04,  // 0x3F4CDF0CEF61A8E9
	-5.38595305356740546715e-04, // 0xBF41A6109C73E0EC
	3.15632070903625950361e-04,  // 0x3F34AF6D6C0EBBF7
	-3.12754168375120860518e-04, // 0xBF347F24ECC38C38
	3.35529192635519073543e-04,  // 0x3F35FD3EE8C2D3F4
}
var _lgamU = [...]float64{
	-7.72156649015328655494e-02, // 0xBFB3C467E37DB0C8
	6.32827064025093366517e-01,  // 0x3FE4401E8B005DFF
	1.45492250137234768737e+00,  // 0x3FF7475CD119BD6F
	9.77717527963372745603e-01,  // 0x3FEF497644EA8450
	2.28963728064692451092e-01,  // 0x3FCD4EAEF6010924
	1.33810918536787660377e-02,  // 0x3F8B678BBF2BAB09
}
var _lgamV = [...]float64{
	1.0,
	2.45597793713041134822e+00, // 0x4003A5D7C2BD619C
	2.12848976379893395361e+00, // 0x40010725A42B18F5
	7.69285150456672783825e-01, // 0x3FE89DFBE45050AF
	1.04222645593369134254e-01, // 0x3FBAAE55D6537C88
	3.21709242282423911810e-03, // 0x3F6A5ABB57D0CF61
}
var _lgamW = [...]float64{
	4.18938533204672725052e-01,  // 0x3FDACFE390C97D69
	8.33333333333329678849e-02,  // 0x3FB555555555553B
	-2.77777777728775536470e-03, // 0xBF66C16C16B02E5C
	7.93650558643019558500e-04,  // 0x3F4A019F98CF38B6
	-5.95187557450339963135e-04, // 0xBF4380CB8C0FE741
	8.36339918996282139126e-04,  // 0x3F4B67BA4CDAD5D1
	-1.63092934096575273989e-03, // 0xBF5AB89D0B9E43E4
}

// Lgamma returns the natural logarithm and sign (-1 or +1) of Gamma(x).
//
// Special cases are:
//	Lgamma(+Inf) = +Inf
//	Lgamma(0) = +Inf
//	Lgamma(-integer) = +Inf
//	Lgamma(-Inf) = -Inf
//	Lgamma(NaN) = NaN
func ( float64) ( float64,  int) {
	const (
		  = 1.461632144968362245
		 = 1 << 52                     // 0x4330000000000000 ~4.5036e+15
		 = 1 << 53                     // 0x4340000000000000 ~9.0072e+15
		 = 1 << 58                     // 0x4390000000000000 ~2.8823e+17
		  = 1.0 / (1 << 70)             // 0x3b90000000000000 ~8.47033e-22
		    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F
		    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
		// Tt = -(tail of Tf)
		 = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
	)
	// special cases
	 = 1
	switch {
	case IsNaN():
		 = 
		return
	case IsInf(, 0):
		 = 
		return
	case  == 0:
		 = Inf(1)
		return
	}

	 := false
	if  < 0 {
		 = -
		 = true
	}

	if  <  { // if |x| < 2**-70, return -log(|x|)
		if  {
			 = -1
		}
		 = -Log()
		return
	}
	var  float64
	if  {
		if  >=  { // |x| >= 2**52, must be -integer
			 = Inf(1)
			return
		}
		 := sinPi()
		if  == 0 {
			 = Inf(1) // -integer
			return
		}
		 = Log(Pi / Abs(*))
		if  < 0 {
			 = -1
		}
	}

	switch {
	case  == 1 ||  == 2: // purge off 1 and 2
		 = 0
		return
	case  < 2: // use lgamma(x) = lgamma(x+1) - log(x)
		var  float64
		var  int
		if  <= 0.9 {
			 = -Log()
			switch {
			case  >= ( - 1 + 0.27): // 0.7316 <= x <=  0.9
				 = 1 - 
				 = 0
			case  >= ( - 1 - 0.27): // 0.2316 <= x < 0.7316
				 =  - ( - 1)
				 = 1
			default: // 0 < x < 0.2316
				 = 
				 = 2
			}
		} else {
			 = 0
			switch {
			case  >= ( + 0.27): // 1.7316 <= x < 2
				 = 2 - 
				 = 0
			case  >= ( - 0.27): // 1.2316 <= x < 1.7316
				 =  - 
				 = 1
			default: // 0.9 < x < 1.2316
				 =  - 1
				 = 2
			}
		}
		switch  {
		case 0:
			 :=  * 
			 := _lgamA[0] + *(_lgamA[2]+*(_lgamA[4]+*(_lgamA[6]+*(_lgamA[8]+*_lgamA[10]))))
			 :=  * (_lgamA[1] + *(+_lgamA[3]+*(_lgamA[5]+*(_lgamA[7]+*(_lgamA[9]+*_lgamA[11])))))
			 := * + 
			 += ( - 0.5*)
		case 1:
			 :=  * 
			 :=  * 
			 := _lgamT[0] + *(_lgamT[3]+*(_lgamT[6]+*(_lgamT[9]+*_lgamT[12]))) // parallel comp
			 := _lgamT[1] + *(_lgamT[4]+*(_lgamT[7]+*(_lgamT[10]+*_lgamT[13])))
			 := _lgamT[2] + *(_lgamT[5]+*(_lgamT[8]+*(_lgamT[11]+*_lgamT[14])))
			 := * - ( - *(+*))
			 += ( + )
		case 2:
			 :=  * (_lgamU[0] + *(_lgamU[1]+*(_lgamU[2]+*(_lgamU[3]+*(_lgamU[4]+*_lgamU[5])))))
			 := 1 + *(_lgamV[1]+*(_lgamV[2]+*(_lgamV[3]+*(_lgamV[4]+*_lgamV[5]))))
			 += (-0.5* + /)
		}
	case  < 8: // 2 <= x < 8
		 := int()
		 :=  - float64()
		 :=  * (_lgamS[0] + *(_lgamS[1]+*(_lgamS[2]+*(_lgamS[3]+*(_lgamS[4]+*(_lgamS[5]+*_lgamS[6]))))))
		 := 1 + *(_lgamR[1]+*(_lgamR[2]+*(_lgamR[3]+*(_lgamR[4]+*(_lgamR[5]+*_lgamR[6])))))
		 = 0.5* + /
		 := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
		switch  {
		case 7:
			 *= ( + 6)
			fallthrough
		case 6:
			 *= ( + 5)
			fallthrough
		case 5:
			 *= ( + 4)
			fallthrough
		case 4:
			 *= ( + 3)
			fallthrough
		case 3:
			 *= ( + 2)
			 += Log()
		}
	case  < : // 8 <= x < 2**58
		 := Log()
		 := 1 / 
		 :=  * 
		 := _lgamW[0] + *(_lgamW[1]+*(_lgamW[2]+*(_lgamW[3]+*(_lgamW[4]+*(_lgamW[5]+*_lgamW[6])))))
		 = (-0.5)*(-1) + 
	default: // 2**58 <= x <= Inf
		 =  * (Log() - 1)
	}
	if  {
		 =  - 
	}
	return
}

// sinPi(x) is a helper function for negative x
func ( float64) float64 {
	const (
		 = 1 << 52 // 0x4330000000000000 ~4.5036e+15
		 = 1 << 53 // 0x4340000000000000 ~9.0072e+15
	)
	if  < 0.25 {
		return -Sin(Pi * )
	}

	// argument reduction
	 := Floor()
	var  int
	if  !=  { // inexact
		 = Mod(, 2)
		 = int( * 4)
	} else {
		if  >=  { // x must be even
			 = 0
			 = 0
		} else {
			if  <  {
				 =  +  // exact
			}
			 = int(1 & Float64bits())
			 = float64()
			 <<= 2
		}
	}
	switch  {
	case 0:
		 = Sin(Pi * )
	case 1, 2:
		 = Cos(Pi * (0.5 - ))
	case 3, 4:
		 = Sin(Pi * (1 - ))
	case 5, 6:
		 = -Cos(Pi * ( - 1.5))
	default:
		 = Sin(Pi * ( - 2))
	}
	return -
}