GLSL: pow vs multiplication for integer exponent - opengl

Question

Which is faster in GLSL:
pow(x, 3.0f);
or
x*x*x;
?
Does exponentiation performance depend on hardware vendor or exponent value?

Answer 1

I wrote a small benchmark, because I was interested in the results.
In my personal case, I was most interested in exponent = 5.
Benchmark code (running in Rem's Studio / LWJGL):
package me.anno.utils.bench
import me.anno.gpu.GFX
import me.anno.gpu.GFX.flat01
import me.anno.gpu.RenderState
import me.anno.gpu.RenderState.useFrame
import me.anno.gpu.framebuffer.Frame
import me.anno.gpu.framebuffer.Framebuffer
import me.anno.gpu.hidden.HiddenOpenGLContext
import me.anno.gpu.shader.Renderer
import me.anno.gpu.shader.Shader
import me.anno.utils.types.Floats.f2
import org.lwjgl.opengl.GL11.*
import java.nio.ByteBuffer
import kotlin.math.roundToInt
fun main() {
fun createShader(code: String) = Shader(
"", null, "" +
"attribute vec2 attr0;\n" +
"void main(){\n" +
" gl_Position = vec4(attr0*2.0-1.0, 0.0, 1.0);\n" +
" uv = attr0;\n" +
"}", "varying vec2 uv;\n", "" +
"void main(){" +
code +
"}"
)
fun repeat(code: String, times: Int): String {
return Array(times) { code }.joinToString("\n")
}
val size = 512
val warmup = 50
val benchmark = 1000
HiddenOpenGLContext.setSize(size, size)
HiddenOpenGLContext.createOpenGL()
val buffer = Framebuffer("", size, size, 1, 1, true, Framebuffer.DepthBufferType.NONE)
println("Power,Multiplications,GFlops-multiplication,GFlops-floats,GFlops-ints,GFlops-power,Speedup")
useFrame(buffer, Renderer.colorRenderer) {
RenderState.blendMode.use(me.anno.gpu.blending.BlendMode.ADD) {
for (power in 2 until 100) {
// to reduce the overhead of other stuff
val repeats = 100
val init = "float x1 = dot(uv, vec2(1.0)),x2,x4,x8,x16,x32,x64;\n"
val end = "gl_FragColor = vec4(x1,x1,x1,x1);\n"
val manualCode = StringBuilder()
for (bit in 1 until 32) {
val p = 1.shl(bit)
val h = 1.shl(bit - 1)
if (power == p) {
manualCode.append("x1=x$h*x$h;")
break
} else if (power > p) {
manualCode.append("x$p=x$h*x$h;")
} else break
}
if (power.and(power - 1) != 0) {
// not a power of two, so the result isn't finished yet
manualCode.append("x1=")
var first = true
for (bit in 0 until 32) {
val p = 1.shl(bit)
if (power.and(p) != 0) {
if (!first) {
manualCode.append('*')
} else first = false
manualCode.append("x$p")
}
}
manualCode.append(";\n")
}
val multiplications = manualCode.count { it == '*' }
// println("$power: $manualCode")
val shaders = listOf(
// manually optimized
createShader(init + repeat(manualCode.toString(), repeats) + end),
// can be optimized
createShader(init + repeat("x1=pow(x1,$power.0);", repeats) + end),
// can be optimized, int as power
createShader(init + repeat("x1=pow(x1,$power);", repeats) + end),
// slightly different, so it can't be optimized
createShader(init + repeat("x1=pow(x1,${power}.01);", repeats) + end),
)
for (shader in shaders) {
shader.use()
}
val pixels = ByteBuffer.allocateDirect(4)
Frame.bind()
glClearColor(0f, 0f, 0f, 1f)
glClear(GL_COLOR_BUFFER_BIT or GL_DEPTH_BUFFER_BIT)
for (i in 0 until warmup) {
for (shader in shaders) {
shader.use()
flat01.draw(shader)
}
}
val flops = DoubleArray(shaders.size)
val avg = 10 // for more stability between runs
for (j in 0 until avg) {
for (index in shaders.indices) {
val shader = shaders[index]
GFX.check()
val t0 = System.nanoTime()
for (i in 0 until benchmark) {
shader.use()
flat01.draw(shader)
}
// synchronize
glReadPixels(0, 0, 1, 1, GL_RGBA, GL_UNSIGNED_BYTE, pixels)
GFX.check()
val t1 = System.nanoTime()
// the first one may be an outlier
if (j > 0) flops[index] += multiplications * repeats.toDouble() * benchmark.toDouble() * size * size / (t1 - t0)
GFX.check()
}
}
for (i in flops.indices) {
flops[i] /= (avg - 1.0)
}
println(
"" +
"$power,$multiplications," +
"${flops[0].roundToInt()}," +
"${flops[1].roundToInt()}," +
"${flops[2].roundToInt()}," +
"${flops[3].roundToInt()}," +
(flops[0] / flops[3]).f2()
)
}
}
}
}
The sampler function is run 9x 512² pixels * 1000 times, and evaluates the function 100 times each.
I run this code on my RX 580, 8GB from Gigabyte, and collected the following results:
Power
#Mult
GFlops*
GFlopsFp
GFlopsInt
GFlopsPow
Speedup
2
1
1246
1429
1447
324
3.84
3
2
2663
2692
2708
651
4.09
4
2
2682
2679
2698
650
4.12
5
3
2766
972
974
973
2.84
6
3
2785
978
974
976
2.85
7
4
2830
1295
1303
1299
2.18
8
3
2783
2792
2809
960
2.90
9
4
2836
1298
1301
1302
2.18
10
4
2833
1291
1302
1298
2.18
11
5
2858
1623
1629
1623
1.76
12
4
2824
1302
1295
1303
2.17
13
5
2866
1628
1624
1626
1.76
14
5
2869
1614
1623
1611
1.78
15
6
2886
1945
1943
1953
1.48
16
4
2821
1305
1300
1305
2.16
17
5
2868
1615
1625
1619
1.77
18
5
2858
1620
1625
1624
1.76
19
6
2890
1949
1946
1949
1.48
20
5
2871
1618
1627
1625
1.77
21
6
2879
1945
1947
1943
1.48
22
6
2886
1944
1949
1952
1.48
23
7
2901
2271
2269
2268
1.28
24
5
2872
1621
1628
1624
1.77
25
6
2886
1942
1943
1942
1.49
26
6
2880
1949
1949
1953
1.47
27
7
2891
2273
2263
2266
1.28
28
6
2883
1949
1946
1953
1.48
29
7
2910
2279
2281
2279
1.28
30
7
2899
2272
2276
2277
1.27
31
8
2906
2598
2595
2596
1.12
32
5
2872
1621
1625
1622
1.77
33
6
2901
1953
1942
1949
1.49
34
6
2895
1948
1939
1944
1.49
35
7
2895
2274
2266
2268
1.28
36
6
2881
1937
1944
1948
1.48
37
7
2894
2277
2270
2280
1.27
38
7
2902
2275
2264
2273
1.28
39
8
2910
2602
2594
2603
1.12
40
6
2877
1945
1947
1945
1.48
41
7
2892
2276
2277
2282
1.27
42
7
2887
2271
2272
2273
1.27
43
8
2912
2599
2606
2599
1.12
44
7
2910
2278
2284
2276
1.28
45
8
2920
2597
2601
2600
1.12
46
8
2920
2600
2601
2590
1.13
47
9
2925
2921
2926
2927
1.00
48
6
2885
1935
1955
1956
1.47
49
7
2901
2271
2279
2288
1.27
50
7
2904
2281
2276
2278
1.27
51
8
2919
2608
2594
2607
1.12
52
7
2902
2282
2270
2273
1.28
53
8
2903
2598
2602
2598
1.12
54
8
2918
2602
2602
2604
1.12
55
9
2932
2927
2924
2936
1.00
56
7
2907
2284
2282
2281
1.27
57
8
2920
2606
2604
2610
1.12
58
8
2913
2593
2597
2587
1.13
59
9
2925
2923
2924
2920
1.00
60
8
2930
2614
2606
2613
1.12
61
9
2932
2946
2946
2947
1.00
62
9
2926
2935
2937
2947
0.99
63
10
2958
3258
3192
3266
0.91
64
6
2902
1957
1956
1959
1.48
65
7
2903
2274
2267
2273
1.28
66
7
2909
2277
2276
2286
1.27
67
8
2908
2602
2606
2599
1.12
68
7
2894
2272
2279
2276
1.27
69
8
2923
2597
2606
2606
1.12
70
8
2910
2596
2599
2600
1.12
71
9
2926
2921
2927
2924
1.00
72
7
2909
2283
2273
2273
1.28
73
8
2909
2602
2602
2599
1.12
74
8
2914
2602
2602
2603
1.12
75
9
2924
2925
2927
2933
1.00
76
8
2904
2608
2602
2601
1.12
77
9
2911
2919
2917
2909
1.00
78
9
2927
2921
2917
2935
1.00
79
10
2929
3241
3246
3246
0.90
80
7
2903
2273
2276
2275
1.28
81
8
2916
2596
2592
2589
1.13
82
8
2913
2600
2597
2598
1.12
83
9
2925
2931
2926
2913
1.00
84
8
2917
2598
2606
2597
1.12
85
9
2920
2916
2918
2927
1.00
86
9
2942
2922
2944
2936
1.00
87
10
2961
3254
3259
3268
0.91
88
8
2934
2607
2608
2612
1.12
89
9
2918
2939
2931
2916
1.00
90
9
2927
2928
2920
2924
1.00
91
10
2940
3253
3252
3246
0.91
92
9
2924
2933
2926
2928
1.00
93
10
2940
3259
3237
3251
0.90
94
10
2928
3247
3247
3264
0.90
95
11
2933
3599
3593
3594
0.82
96
7
2883
2282
2268
2269
1.27
97
8
2911
2602
2595
2600
1.12
98
8
2896
2588
2591
2587
1.12
99
9
2924
2939
2936
2938
1.00
As you can see, a power() call takes exactly as long as 9 multiplication instructions. Therefore every manual rewriting of a power with less than 9 multiplications is faster.
Only the cases 2, 3, 4, and 8 are optimized by my driver. The optimization is independent of whether you use the .0 suffix for the exponent.
In the case of exponent = 2, my implementation seems to have lower performance than the driver. I am not sure, why.
The speedup is the manual implementation compared to pow(x,exponent+0.01), which cannot be optimized by the compiler.
Because the multiplications and the speedup align so perfectly, I created a graph to show the relationship. This relationship kind of shows that my benchmark is trustworthy :).
Operating System: Windows 10 Personal
GPU: RX 580 8GB from Gigabyte
Processor: Ryzen 5 2600
Memory: 16 GB DDR4 3200
GPU Driver: 21.6.1 from 17th June 2021
LWJGL: Version 3.2.3 build 13

Answer 2

While this can definitely be hardware/vendor/compiler dependent, advanced mathematical functions like pow() tend to be considerably more expensive than basic operations.
The best approach is of course to try both, and benchmark. But if there is a simple replacement for an advanced mathematical functions, I don't think you can go very wrong by using it.
If you write pow(x, 3.0), the best you can probably hope for is that the compiler will recognize the special case, and expand it. But why take the risk, if the replacement is just as short and easy to read? C/C++ compilers don't always replace pow(x, 2.0) by a simple multiplication, so I wouldn't necessarily count on all GLSL compilers to do that.