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 e117158..ae2c0c4 100644
--- a/src/pkg/math/gamma.go
+++ b/src/pkg/math/gamma.go
@@ -63,7 +63,7 @@
// 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 @@
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 @@
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 @@
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 @@
}
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 8f6d7b9..e2bad69 100644
--- a/src/pkg/math/lgamma.go
+++ b/src/pkg/math/lgamma.go
@@ -88,6 +88,81 @@
//
//
+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 @@
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 @@
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 @@
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)
