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

diff --git a/src/pkg/math/gamma.go b/src/pkg/math/gamma.go
index e117158fee2e28822e1abee8803c3117a865f5d0..ae2c0c418ab7b04cb3a03adfa7c2e4cae200eb07 100644 (file)
--- a/src/pkg/math/gamma.go
+++ b/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:

diff --git a/src/pkg/math/lgamma.go b/src/pkg/math/lgamma.go
index 8f6d7b99fc5e98b82ad262cbb8d058359e40a414..e2bad69dc03d0f8d9fa2245419122f3121f37846 100644 (file)
--- a/src/pkg/math/lgamma.go
+++ b/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)