math: faster Lgamma

Converting from polynomial constants to counted array speeds up Lgamma from 51.3 to 37.7 ns/op. Variables renamed in Gamma to avoid overlap in Lgamma.

R=rsc, golang-dev
CC=golang-dev
https://golang.org/cl/5359045

src/pkg/math/gamma.go

@@ -63,7 +63,7 @@ package math
 //   Stephen L. Moshier
 //   moshier@na-net.ornl.gov
 
-var _P = [...]float64{
+var _gamP = [...]float64{
 	1.60119522476751861407e-04,
 	1.19135147006586384913e-03,
 	1.04213797561761569935e-02,
@@ -72,7 +72,7 @@ var _P = [...]float64{
 	4.94214826801497100753e-01,
 	9.99999999999999996796e-01,
 }
-var _Q = [...]float64{
+var _gamQ = [...]float64{
 	-2.31581873324120129819e-05,
 	5.39605580493303397842e-04,
 	-4.45641913851797240494e-03,
@@ -82,7 +82,7 @@ var _Q = [...]float64{
 	7.14304917030273074085e-02,
 	1.00000000000000000320e+00,
 }
-var _S = [...]float64{
+var _gamS = [...]float64{
 	7.87311395793093628397e-04,
 	-2.29549961613378126380e-04,
 	-2.68132617805781232825e-03,
@@ -98,7 +98,7 @@ func stirling(x float64) float64 {
 		MaxStirling = 143.01608
 	)
 	w := 1 / x
-	w = 1 + w*((((_S[0]*w+_S[1])*w+_S[2])*w+_S[3])*w+_S[4])
+	w = 1 + w*((((_gamS[0]*w+_gamS[1])*w+_gamS[2])*w+_gamS[3])*w+_gamS[4])
 	y := Exp(x)
 	if x > MaxStirling { // avoid Pow() overflow
 		v := Pow(x, 0.5*x-0.25)
@@ -176,8 +176,8 @@ func Gamma(x float64) float64 {
 	}
 
 	x = x - 2
-	p = (((((x*_P[0]+_P[1])*x+_P[2])*x+_P[3])*x+_P[4])*x+_P[5])*x + _P[6]
-	q = ((((((x*_Q[0]+_Q[1])*x+_Q[2])*x+_Q[3])*x+_Q[4])*x+_Q[5])*x+_Q[6])*x + _Q[7]
+	p = (((((x*_gamP[0]+_gamP[1])*x+_gamP[2])*x+_gamP[3])*x+_gamP[4])*x+_gamP[5])*x + _gamP[6]
+	q = ((((((x*_gamQ[0]+_gamQ[1])*x+_gamQ[2])*x+_gamQ[3])*x+_gamQ[4])*x+_gamQ[5])*x+_gamQ[6])*x + _gamQ[7]
 	return z * p / q
 
 small:

src/pkg/math/lgamma.go

@@ -88,6 +88,81 @@ package math
 //
 //
 
+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:
@@ -103,68 +178,10 @@ func Lgamma(x float64) (lgamma float64, sign int) {
 		Two53 = 1 << 53                     // 0x4340000000000000 ~9.0072e+15
 		Two58 = 1 << 58                     // 0x4390000000000000 ~2.8823e+17
 		Tiny  = 1.0 / (1 << 70)             // 0x3b90000000000000 ~8.47033e-22
-		A0    = 7.72156649015328655494e-02  // 0x3FB3C467E37DB0C8
-		A1    = 3.22467033424113591611e-01  // 0x3FD4A34CC4A60FAD
-		A2    = 6.73523010531292681824e-02  // 0x3FB13E001A5562A7
-		A3    = 2.05808084325167332806e-02  // 0x3F951322AC92547B
-		A4    = 7.38555086081402883957e-03  // 0x3F7E404FB68FEFE8
-		A5    = 2.89051383673415629091e-03  // 0x3F67ADD8CCB7926B
-		A6    = 1.19270763183362067845e-03  // 0x3F538A94116F3F5D
-		A7    = 5.10069792153511336608e-04  // 0x3F40B6C689B99C00
-		A8    = 2.20862790713908385557e-04  // 0x3F2CF2ECED10E54D
-		A9    = 1.08011567247583939954e-04  // 0x3F1C5088987DFB07
-		A10   = 2.52144565451257326939e-05  // 0x3EFA7074428CFA52
-		A11   = 4.48640949618915160150e-05  // 0x3F07858E90A45837
 		Tc    = 1.46163214496836224576e+00  // 0x3FF762D86356BE3F
 		Tf    = -1.21486290535849611461e-01 // 0xBFBF19B9BCC38A42
 		// Tt = -(tail of Tf)
-		Tt  = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
-		T0  = 4.83836122723810047042e-01  // 0x3FDEF72BC8EE38A2
-		T1  = -1.47587722994593911752e-01 // 0xBFC2E4278DC6C509
-		T2  = 6.46249402391333854778e-02  // 0x3FB08B4294D5419B
-		T3  = -3.27885410759859649565e-02 // 0xBFA0C9A8DF35B713
-		T4  = 1.79706750811820387126e-02  // 0x3F9266E7970AF9EC
-		T5  = -1.03142241298341437450e-02 // 0xBF851F9FBA91EC6A
-		T6  = 6.10053870246291332635e-03  // 0x3F78FCE0E370E344
-		T7  = -3.68452016781138256760e-03 // 0xBF6E2EFFB3E914D7
-		T8  = 2.25964780900612472250e-03  // 0x3F6282D32E15C915
-		T9  = -1.40346469989232843813e-03 // 0xBF56FE8EBF2D1AF1
-		T10 = 8.81081882437654011382e-04  // 0x3F4CDF0CEF61A8E9
-		T11 = -5.38595305356740546715e-04 // 0xBF41A6109C73E0EC
-		T12 = 3.15632070903625950361e-04  // 0x3F34AF6D6C0EBBF7
-		T13 = -3.12754168375120860518e-04 // 0xBF347F24ECC38C38
-		T14 = 3.35529192635519073543e-04  // 0x3F35FD3EE8C2D3F4
-		U0  = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
-		U1  = 6.32827064025093366517e-01  // 0x3FE4401E8B005DFF
-		U2  = 1.45492250137234768737e+00  // 0x3FF7475CD119BD6F
-		U3  = 9.77717527963372745603e-01  // 0x3FEF497644EA8450
-		U4  = 2.28963728064692451092e-01  // 0x3FCD4EAEF6010924
-		U5  = 1.33810918536787660377e-02  // 0x3F8B678BBF2BAB09
-		V1  = 2.45597793713041134822e+00  // 0x4003A5D7C2BD619C
-		V2  = 2.12848976379893395361e+00  // 0x40010725A42B18F5
-		V3  = 7.69285150456672783825e-01  // 0x3FE89DFBE45050AF
-		V4  = 1.04222645593369134254e-01  // 0x3FBAAE55D6537C88
-		V5  = 3.21709242282423911810e-03  // 0x3F6A5ABB57D0CF61
-		S0  = -7.72156649015328655494e-02 // 0xBFB3C467E37DB0C8
-		S1  = 2.14982415960608852501e-01  // 0x3FCB848B36E20878
-		S2  = 3.25778796408930981787e-01  // 0x3FD4D98F4F139F59
-		S3  = 1.46350472652464452805e-01  // 0x3FC2BB9CBEE5F2F7
-		S4  = 2.66422703033638609560e-02  // 0x3F9B481C7E939961
-		S5  = 1.84028451407337715652e-03  // 0x3F5E26B67368F239
-		S6  = 3.19475326584100867617e-05  // 0x3F00BFECDD17E945
-		R1  = 1.39200533467621045958e+00  // 0x3FF645A762C4AB74
-		R2  = 7.21935547567138069525e-01  // 0x3FE71A1893D3DCDC
-		R3  = 1.71933865632803078993e-01  // 0x3FC601EDCCFBDF27
-		R4  = 1.86459191715652901344e-02  // 0x3F9317EA742ED475
-		R5  = 7.77942496381893596434e-04  // 0x3F497DDACA41A95B
-		R6  = 7.32668430744625636189e-06  // 0x3EDEBAF7A5B38140
-		W0  = 4.18938533204672725052e-01  // 0x3FDACFE390C97D69
-		W1  = 8.33333333333329678849e-02  // 0x3FB555555555553B
-		W2  = -2.77777777728775536470e-03 // 0xBF66C16C16B02E5C
-		W3  = 7.93650558643019558500e-04  // 0x3F4A019F98CF38B6
-		W4  = -5.95187557450339963135e-04 // 0xBF4380CB8C0FE741
-		W5  = 8.36339918996282139126e-04  // 0x3F4B67BA4CDAD5D1
-		W6  = -1.63092934096575273989e-03 // 0xBF5AB89D0B9E43E4
+		Tt = -3.63867699703950536541e-18 // 0xBC50C7CAA48A971F
 	)
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
@@ -249,28 +266,28 @@ func Lgamma(x float64) (lgamma float64, sign int) {
 		switch i {
 		case 0:
 			z := y * y
-			p1 := A0 + z*(A2+z*(A4+z*(A6+z*(A8+z*A10))))
-			p2 := z * (A1 + z*(A3+z*(A5+z*(A7+z*(A9+z*A11)))))
+			p1 := _lgamA[0] + z*(_lgamA[2]+z*(_lgamA[4]+z*(_lgamA[6]+z*(_lgamA[8]+z*_lgamA[10]))))
+			p2 := z * (_lgamA[1] + z*(+_lgamA[3]+z*(_lgamA[5]+z*(_lgamA[7]+z*(_lgamA[9]+z*_lgamA[11])))))
 			p := y*p1 + p2
 			lgamma += (p - 0.5*y)
 		case 1:
 			z := y * y
 			w := z * y
-			p1 := T0 + w*(T3+w*(T6+w*(T9+w*T12))) // parallel comp
-			p2 := T1 + w*(T4+w*(T7+w*(T10+w*T13)))
-			p3 := T2 + w*(T5+w*(T8+w*(T11+w*T14)))
+			p1 := _lgamT[0] + w*(_lgamT[3]+w*(_lgamT[6]+w*(_lgamT[9]+w*_lgamT[12]))) // parallel comp
+			p2 := _lgamT[1] + w*(_lgamT[4]+w*(_lgamT[7]+w*(_lgamT[10]+w*_lgamT[13])))
+			p3 := _lgamT[2] + w*(_lgamT[5]+w*(_lgamT[8]+w*(_lgamT[11]+w*_lgamT[14])))
 			p := z*p1 - (Tt - w*(p2+y*p3))
 			lgamma += (Tf + p)
 		case 2:
-			p1 := y * (U0 + y*(U1+y*(U2+y*(U3+y*(U4+y*U5)))))
-			p2 := 1 + y*(V1+y*(V2+y*(V3+y*(V4+y*V5))))
+			p1 := y * (_lgamU[0] + y*(_lgamU[1]+y*(_lgamU[2]+y*(_lgamU[3]+y*(_lgamU[4]+y*_lgamU[5])))))
+			p2 := 1 + y*(_lgamV[1]+y*(_lgamV[2]+y*(_lgamV[3]+y*(_lgamV[4]+y*_lgamV[5]))))
 			lgamma += (-0.5*y + p1/p2)
 		}
 	case x < 8: // 2 <= x < 8
 		i := int(x)
 		y := x - float64(i)
-		p := y * (S0 + y*(S1+y*(S2+y*(S3+y*(S4+y*(S5+y*S6))))))
-		q := 1 + y*(R1+y*(R2+y*(R3+y*(R4+y*(R5+y*R6)))))
+		p := y * (_lgamS[0] + y*(_lgamS[1]+y*(_lgamS[2]+y*(_lgamS[3]+y*(_lgamS[4]+y*(_lgamS[5]+y*_lgamS[6]))))))
+		q := 1 + y*(_lgamR[1]+y*(_lgamR[2]+y*(_lgamR[3]+y*(_lgamR[4]+y*(_lgamR[5]+y*_lgamR[6])))))
 		lgamma = 0.5*y + p/q
 		z := 1.0 // Lgamma(1+s) = Log(s) + Lgamma(s)
 		switch i {
@@ -294,7 +311,7 @@ func Lgamma(x float64) (lgamma float64, sign int) {
 		t := Log(x)
 		z := 1 / x
 		y := z * z
-		w := W0 + z*(W1+y*(W2+y*(W3+y*(W4+y*(W5+y*W6)))))
+		w := _lgamW[0] + z*(_lgamW[1]+y*(_lgamW[2]+y*(_lgamW[3]+y*(_lgamW[4]+y*(_lgamW[5]+y*_lgamW[6])))))
 		lgamma = (x-0.5)*(t-1) + w
 	default: // 2**58 <= x <= Inf
 		lgamma = x * (Log(x) - 1)