Lookup Tables

Objective

To discuss lookup tables and how to use them to sacrifice storage space to increase computation time.

What Are Lookup Tables

Lookup tables are static arrays that sacrifices memory storage in place of a simple array index lookup of precalculated values. In some examples, a lookup table is not meant to speed a process, but simply an elegant solution to a problem.

Lets look at some examples to see why these are useful.

Why Use Lookup Tables

Simple Example: Convert Potentiometer Voltage to Angle

Lets make some assumptions about the system first:

  1. Using an 8-bit ADC
  2. Potentiometer is linear
  3. Potentiometer sweep angle is 180 degrees
  4. Potentiometer all the way left is 0 deg and 0V
  5. Potentiometer all the way right (180 deg) is ADC Reference Voltage
  6. Using a processor that does NOT have a FPU (Floating Point arithmetic Unit) like the Arm Cortex M3 we use in the LPC1756.
double potADCToDegrees(uint8_t adc)
{
	return ((double)(adc))*(270/256);
}

Code Block 1. Without Lookup

const double potentiometer_angles[256] =
{
//  [ADC] = Angle
	[0]		= 0.0,
	[1] 	= 1.0546875,
	[2] 	= 2.109375,
	[3] 	= 3.1640625,
	[4] 	= 4.21875,
	[5] 	= 5.2734375,
	[6] 	= 6.328125,
	[7] 	= 7.3828125,
	[8] 	= 8.4375,
	[9] 	= 9.4921875,
	[10] 	= 10.546875,
	[11] 	= 11.6015625,
	[12] 	= 12.65625,
	[13] 	= 13.7109375,
	[14] 	= 14.765625,
	[15] 	= 15.8203125,
	[16] 	= 16.875,
	[17] 	= 17.9296875,
	[18] 	= 18.984375,
	[19] 	= 20.0390625,
	[20] 	= 21.09375,
	[21] 	= 22.1484375,
	[22] 	= 23.203125,
	[23] 	= 24.2578125,
	[24] 	= 25.3125,
	[25] 	= 26.3671875,
	[26] 	= 27.421875,
	[27] 	= 28.4765625,
	[28] 	= 29.53125,
	[29] 	= 30.5859375,
	[30] 	= 31.640625,
	[31] 	= 32.6953125,
	[32] 	= 33.75,
	[33] 	= 34.8046875,
	[34] 	= 35.859375,
	[35] 	= 36.9140625,
	[36] 	= 37.96875,
	[37] 	= 39.0234375,
	[38] 	= 40.078125,
	[39] 	= 41.1328125,
	[40] 	= 42.1875,
	[41] 	= 43.2421875,
	[42] 	= 44.296875,
	[43] 	= 45.3515625,
	[44] 	= 46.40625,
	[45] 	= 47.4609375,
	[46] 	= 48.515625,
	[47] 	= 49.5703125,
	[48] 	= 50.625,
	[49] 	= 51.6796875,
	[50] 	= 52.734375,
	[51] 	= 53.7890625,
	[52] 	= 54.84375,
	[53] 	= 55.8984375,
	[54] 	= 56.953125,
	[55] 	= 58.0078125,
	[56] 	= 59.0625,
	[57] 	= 60.1171875,
	[58] 	= 61.171875,
	[59] 	= 62.2265625,
	[60] 	= 63.28125,
	[61] 	= 64.3359375,
	[62] 	= 65.390625,
	[63] 	= 66.4453125,
	[64] 	= 67.5,
	[65] 	= 68.5546875,
	[66] 	= 69.609375,
	[67] 	= 70.6640625,
	[68] 	= 71.71875,
	[69] 	= 72.7734375,
	[70] 	= 73.828125,
	[71] 	= 74.8828125,
	[72] 	= 75.9375,
	[73] 	= 76.9921875,
	[74] 	= 78.046875,
	[75] 	= 79.1015625,
	[76] 	= 80.15625,
	[77] 	= 81.2109375,
	[78] 	= 82.265625,
	[79] 	= 83.3203125,
	[80] 	= 84.375,
	[81] 	= 85.4296875,
	[82] 	= 86.484375,
	[83] 	= 87.5390625,
	[84] 	= 88.59375,
	[85] 	= 89.6484375,
	[86] 	= 90.703125,
	[87] 	= 91.7578125,
	[88] 	= 92.8125,
	[89] 	= 93.8671875,
	[90] 	= 94.921875,
	[91] 	= 95.9765625,
	[92] 	= 97.03125,
	[93] 	= 98.0859375,
	[94] 	= 99.140625,
	[95] 	= 100.1953125,
	[96] 	= 101.25,
	[97] 	= 102.3046875,
	[98] 	= 103.359375,
	[99] 	= 104.4140625,
	[100] 	= 105.46875,
	[101] 	= 106.5234375,
	[102] 	= 107.578125,
	[103] 	= 108.6328125,
	[104] 	= 109.6875,
	[105] 	= 110.7421875,
	[106] 	= 111.796875,
	[107] 	= 112.8515625,
	[108] 	= 113.90625,
	[109] 	= 114.9609375,
	[110] 	= 116.015625,
	[111] 	= 117.0703125,
	[112] 	= 118.125,
	[113] 	= 119.1796875,
	[114] 	= 120.234375,
	[115] 	= 121.2890625,
	[116] 	= 122.34375,
	[117] 	= 123.3984375,
	[118] 	= 124.453125,
	[119] 	= 125.5078125,
	[120] 	= 126.5625,
	[121] 	= 127.6171875,
	[122] 	= 128.671875,
	[123] 	= 129.7265625,
	[124] 	= 130.78125,
	[125] 	= 131.8359375,
	[126] 	= 132.890625,
	[127] 	= 133.9453125,
	[128] 	= 135,
	[129] 	= 136.0546875,
	[130] 	= 137.109375,
	[131] 	= 138.1640625,
	[132] 	= 139.21875,
	[133] 	= 140.2734375,
	[134] 	= 141.328125,
	[135] 	= 142.3828125,
	[136] 	= 143.4375,
	[137] 	= 144.4921875,
	[138] 	= 145.546875,
	[139] 	= 146.6015625,
	[140] 	= 147.65625,
	[141] 	= 148.7109375,
	[142] 	= 149.765625,
	[143] 	= 150.8203125,
	[144] 	= 151.875,
	[145] 	= 152.9296875,
	[146] 	= 153.984375,
	[147] 	= 155.0390625,
	[148] 	= 156.09375,
	[149] 	= 157.1484375,
	[150] 	= 158.203125,
	[151] 	= 159.2578125,
	[152] 	= 160.3125,
	[153] 	= 161.3671875,
	[154] 	= 162.421875,
	[155] 	= 163.4765625,
	[156] 	= 164.53125,
	[157] 	= 165.5859375,
	[158] 	= 166.640625,
	[159] 	= 167.6953125,
	[160] 	= 168.75,
	[161] 	= 169.8046875,
	[162] 	= 170.859375,
	[163] 	= 171.9140625,
	[164] 	= 172.96875,
	[165] 	= 174.0234375,
	[166] 	= 175.078125,
	[167] 	= 176.1328125,
	[168] 	= 177.1875,
	[169] 	= 178.2421875,
	[170] 	= 179.296875,
	[171] 	= 180.3515625,
	[172] 	= 181.40625,
	[173] 	= 182.4609375,
	[174] 	= 183.515625,
	[175] 	= 184.5703125,
	[176] 	= 185.625,
	[177] 	= 186.6796875,
	[178] 	= 187.734375,
	[179] 	= 188.7890625,
	[180] 	= 189.84375,
	[181] 	= 190.8984375,
	[182] 	= 191.953125,
	[183] 	= 193.0078125,
	[184] 	= 194.0625,
	[185] 	= 195.1171875,
	[186] 	= 196.171875,
	[187] 	= 197.2265625,
	[188] 	= 198.28125,
	[189] 	= 199.3359375,
	[190] 	= 200.390625,
	[191] 	= 201.4453125,
	[192] 	= 202.5,
	[193] 	= 203.5546875,
	[194] 	= 204.609375,
	[195] 	= 205.6640625,
	[196] 	= 206.71875,
	[197] 	= 207.7734375,
	[198] 	= 208.828125,
	[199] 	= 209.8828125,
	[200] 	= 210.9375,
	[201] 	= 211.9921875,
	[202] 	= 213.046875,
	[203] 	= 214.1015625,
	[204] 	= 215.15625,
	[205] 	= 216.2109375,
	[206] 	= 217.265625,
	[207] 	= 218.3203125,
	[208] 	= 219.375,
	[209] 	= 220.4296875,
	[210] 	= 221.484375,
	[211] 	= 222.5390625,
	[212] 	= 223.59375,
	[213] 	= 224.6484375,
	[214] 	= 225.703125,
	[215] 	= 226.7578125,
	[216] 	= 227.8125,
	[217] 	= 228.8671875,
	[218] 	= 229.921875,
	[219] 	= 230.9765625,
	[220] 	= 232.03125,
	[221] 	= 233.0859375,
	[222] 	= 234.140625,
	[223] 	= 235.1953125,
	[224] 	= 236.25,
	[225] 	= 237.3046875,
	[226] 	= 238.359375,
	[227] 	= 239.4140625,
	[228] 	= 240.46875,
	[229] 	= 241.5234375,
	[230] 	= 242.578125,
	[231] 	= 243.6328125,
	[232] 	= 244.6875,
	[233] 	= 245.7421875,
	[234] 	= 246.796875,
	[235] 	= 247.8515625,
	[236] 	= 248.90625,
	[237] 	= 249.9609375,
	[238] 	= 251.015625,
	[239] 	= 252.0703125,
	[240] 	= 253.125,
	[241] 	= 254.1796875,
	[242] 	= 255.234375,
	[243] 	= 256.2890625,
	[244] 	= 257.34375,
	[245] 	= 258.3984375,
	[246] 	= 259.453125,
	[247] 	= 260.5078125,
	[248] 	= 261.5625,
	[249] 	= 262.6171875,
	[250] 	= 263.671875,
	[251] 	= 264.7265625,
	[252] 	= 265.78125,
	[253] 	= 266.8359375,
	[254] 	= 267.890625,
	[255] 	= 268.9453125,
	[256] 	= 270
};

inline double potADCToDegrees(uint8_t adc)
{
	return potentiometer_angles[adc];
}

Code Block 2. With Lookup

With the two examples, it may seem trivial since the WITHOUT case is only "really" doing one calculation, mulitplying the uint8_t with (270/256) since the compiler will most likely optimize this value to its result. But if you take a look at the assembly, the results may shock you.

Look up Table Disassembly
00016e08 <main>:
main():
/var/www/html/SJSU-Dev/firmware/Experiements/L5_Application/main.cpp:322
    [254]   = 268.9411765,
    [255]   = 270
};

int main(void)
{
   16e08:	b082      	sub	sp, #8
/var/www/html/SJSU-Dev/firmware/Experiements/L5_Application/main.cpp:323
    volatile double a = potentiometer_angles[15];
   16e0a:	a303      	add	r3, pc, #12	; (adr r3, 16e18 <main+0x10>)
   16e0c:	e9d3 2300 	ldrd	r2, r3, [r3]
   16e10:	e9cd 2300 	strd	r2, r3, [sp]
   16e14:	e7fe      	b.n	16e14 <main+0xc>
   16e16:	bf00      	nop
   16e18:	c3b9a8ae 	.word	0xc3b9a8ae
   16e1c:	402fc3c3 	.word	0x402fc3c3

Code Block 3. Dissassembly of Look up Table

Looks about right. You can see at 16e0a the software is retrieving  data from the lookup table, and then it is loading it into the double which is on the stack.

Double Floating Point Disassembly
00017c64 <__adddf3>:
__aeabi_dadd():
   17c64:	b530      	push	{r4, r5, lr}
   17c66:	ea4f 0441 	mov.w	r4, r1, lsl #1
   17c6a:	ea4f 0543 	mov.w	r5, r3, lsl #1
   17c6e:	ea94 0f05 	teq	r4, r5
   17c72:	bf08      	it	eq
   17c74:	ea90 0f02 	teqeq	r0, r2
   17c78:	bf1f      	itttt	ne
   17c7a:	ea54 0c00 	orrsne.w	ip, r4, r0
   17c7e:	ea55 0c02 	orrsne.w	ip, r5, r2
   17c82:	ea7f 5c64 	mvnsne.w	ip, r4, asr #21
   17c86:	ea7f 5c65 	mvnsne.w	ip, r5, asr #21
   17c8a:	f000 80e2 	beq.w	17e52 <__adddf3+0x1ee>
   17c8e:	ea4f 5454 	mov.w	r4, r4, lsr #21
   17c92:	ebd4 5555 	rsbs	r5, r4, r5, lsr #21
   17c96:	bfb8      	it	lt
   17c98:	426d      	neglt	r5, r5
   17c9a:	dd0c      	ble.n	17cb6 <__adddf3+0x52>
   17c9c:	442c      	add	r4, r5
   17c9e:	ea80 0202 	eor.w	r2, r0, r2
   17ca2:	ea81 0303 	eor.w	r3, r1, r3
   17ca6:	ea82 0000 	eor.w	r0, r2, r0
   17caa:	ea83 0101 	eor.w	r1, r3, r1
   17cae:	ea80 0202 	eor.w	r2, r0, r2
   17cb2:	ea81 0303 	eor.w	r3, r1, r3
   17cb6:	2d36      	cmp	r5, #54	; 0x36
   17cb8:	bf88      	it	hi
   17cba:	bd30      	pophi	{r4, r5, pc}
   17cbc:	f011 4f00 	tst.w	r1, #2147483648	; 0x80000000
   17cc0:	ea4f 3101 	mov.w	r1, r1, lsl #12
   17cc4:	f44f 1c80 	mov.w	ip, #1048576	; 0x100000
   17cc8:	ea4c 3111 	orr.w	r1, ip, r1, lsr #12
   17ccc:	d002      	beq.n	17cd4 <__adddf3+0x70>
   17cce:	4240      	negs	r0, r0
   17cd0:	eb61 0141 	sbc.w	r1, r1, r1, lsl #1
   17cd4:	f013 4f00 	tst.w	r3, #2147483648	; 0x80000000
   17cd8:	ea4f 3303 	mov.w	r3, r3, lsl #12
   17cdc:	ea4c 3313 	orr.w	r3, ip, r3, lsr #12
   17ce0:	d002      	beq.n	17ce8 <__adddf3+0x84>
   17ce2:	4252      	negs	r2, r2
   17ce4:	eb63 0343 	sbc.w	r3, r3, r3, lsl #1
   17ce8:	ea94 0f05 	teq	r4, r5
   17cec:	f000 80a7 	beq.w	17e3e <__adddf3+0x1da>
   17cf0:	f1a4 0401 	sub.w	r4, r4, #1
   17cf4:	f1d5 0e20 	rsbs	lr, r5, #32
   17cf8:	db0d      	blt.n	17d16 <__adddf3+0xb2>
   17cfa:	fa02 fc0e 	lsl.w	ip, r2, lr
   17cfe:	fa22 f205 	lsr.w	r2, r2, r5
   17d02:	1880      	adds	r0, r0, r2
   17d04:	f141 0100 	adc.w	r1, r1, #0
   17d08:	fa03 f20e 	lsl.w	r2, r3, lr
   17d0c:	1880      	adds	r0, r0, r2
   17d0e:	fa43 f305 	asr.w	r3, r3, r5
   17d12:	4159      	adcs	r1, r3
   17d14:	e00e      	b.n	17d34 <__adddf3+0xd0>
   17d16:	f1a5 0520 	sub.w	r5, r5, #32
   17d1a:	f10e 0e20 	add.w	lr, lr, #32
   17d1e:	2a01      	cmp	r2, #1
   17d20:	fa03 fc0e 	lsl.w	ip, r3, lr
   17d24:	bf28      	it	cs
   17d26:	f04c 0c02 	orrcs.w	ip, ip, #2
   17d2a:	fa43 f305 	asr.w	r3, r3, r5
   17d2e:	18c0      	adds	r0, r0, r3
   17d30:	eb51 71e3 	adcs.w	r1, r1, r3, asr #31
   17d34:	f001 4500 	and.w	r5, r1, #2147483648	; 0x80000000
   17d38:	d507      	bpl.n	17d4a <__adddf3+0xe6>
   17d3a:	f04f 0e00 	mov.w	lr, #0
   17d3e:	f1dc 0c00 	rsbs	ip, ip, #0
   17d42:	eb7e 0000 	sbcs.w	r0, lr, r0
   17d46:	eb6e 0101 	sbc.w	r1, lr, r1
   17d4a:	f5b1 1f80 	cmp.w	r1, #1048576	; 0x100000
   17d4e:	d31b      	bcc.n	17d88 <__adddf3+0x124>
   17d50:	f5b1 1f00 	cmp.w	r1, #2097152	; 0x200000
   17d54:	d30c      	bcc.n	17d70 <__adddf3+0x10c>
   17d56:	0849      	lsrs	r1, r1, #1
   17d58:	ea5f 0030 	movs.w	r0, r0, rrx
   17d5c:	ea4f 0c3c 	mov.w	ip, ip, rrx
   17d60:	f104 0401 	add.w	r4, r4, #1
   17d64:	ea4f 5244 	mov.w	r2, r4, lsl #21
   17d68:	f512 0f80 	cmn.w	r2, #4194304	; 0x400000
   17d6c:	f080 809a 	bcs.w	17ea4 <__adddf3+0x240>
   17d70:	f1bc 4f00 	cmp.w	ip, #2147483648	; 0x80000000
   17d74:	bf08      	it	eq
   17d76:	ea5f 0c50 	movseq.w	ip, r0, lsr #1
   17d7a:	f150 0000 	adcs.w	r0, r0, #0
   17d7e:	eb41 5104 	adc.w	r1, r1, r4, lsl #20
   17d82:	ea41 0105 	orr.w	r1, r1, r5
   17d86:	bd30      	pop	{r4, r5, pc}
   17d88:	ea5f 0c4c 	movs.w	ip, ip, lsl #1
   17d8c:	4140      	adcs	r0, r0
   17d8e:	eb41 0101 	adc.w	r1, r1, r1
   17d92:	f411 1f80 	tst.w	r1, #1048576	; 0x100000
   17d96:	f1a4 0401 	sub.w	r4, r4, #1
   17d9a:	d1e9      	bne.n	17d70 <__adddf3+0x10c>
   17d9c:	f091 0f00 	teq	r1, #0
   17da0:	bf04      	itt	eq
   17da2:	4601      	moveq	r1, r0
   17da4:	2000      	moveq	r0, #0
   17da6:	fab1 f381 	clz	r3, r1
   17daa:	bf08      	it	eq
   17dac:	3320      	addeq	r3, #32
   17dae:	f1a3 030b 	sub.w	r3, r3, #11
   17db2:	f1b3 0220 	subs.w	r2, r3, #32
   17db6:	da0c      	bge.n	17dd2 <__adddf3+0x16e>
   17db8:	320c      	adds	r2, #12
   17dba:	dd08      	ble.n	17dce <__adddf3+0x16a>
   17dbc:	f102 0c14 	add.w	ip, r2, #20
   17dc0:	f1c2 020c 	rsb	r2, r2, #12
   17dc4:	fa01 f00c 	lsl.w	r0, r1, ip
   17dc8:	fa21 f102 	lsr.w	r1, r1, r2
   17dcc:	e00c      	b.n	17de8 <__adddf3+0x184>
   17dce:	f102 0214 	add.w	r2, r2, #20
   17dd2:	bfd8      	it	le
   17dd4:	f1c2 0c20 	rsble	ip, r2, #32
   17dd8:	fa01 f102 	lsl.w	r1, r1, r2
   17ddc:	fa20 fc0c 	lsr.w	ip, r0, ip
   17de0:	bfdc      	itt	le
   17de2:	ea41 010c 	orrle.w	r1, r1, ip
   17de6:	4090      	lslle	r0, r2
   17de8:	1ae4      	subs	r4, r4, r3
   17dea:	bfa2      	ittt	ge
   17dec:	eb01 5104 	addge.w	r1, r1, r4, lsl #20
   17df0:	4329      	orrge	r1, r5
   17df2:	bd30      	popge	{r4, r5, pc}
   17df4:	ea6f 0404 	mvn.w	r4, r4
   17df8:	3c1f      	subs	r4, #31
   17dfa:	da1c      	bge.n	17e36 <__adddf3+0x1d2>
   17dfc:	340c      	adds	r4, #12
   17dfe:	dc0e      	bgt.n	17e1e <__adddf3+0x1ba>
   17e00:	f104 0414 	add.w	r4, r4, #20
   17e04:	f1c4 0220 	rsb	r2, r4, #32
   17e08:	fa20 f004 	lsr.w	r0, r0, r4
   17e0c:	fa01 f302 	lsl.w	r3, r1, r2
   17e10:	ea40 0003 	orr.w	r0, r0, r3
   17e14:	fa21 f304 	lsr.w	r3, r1, r4
   17e18:	ea45 0103 	orr.w	r1, r5, r3
   17e1c:	bd30      	pop	{r4, r5, pc}
   17e1e:	f1c4 040c 	rsb	r4, r4, #12
   17e22:	f1c4 0220 	rsb	r2, r4, #32
   17e26:	fa20 f002 	lsr.w	r0, r0, r2
   17e2a:	fa01 f304 	lsl.w	r3, r1, r4
   17e2e:	ea40 0003 	orr.w	r0, r0, r3
   17e32:	4629      	mov	r1, r5
   17e34:	bd30      	pop	{r4, r5, pc}
   17e36:	fa21 f004 	lsr.w	r0, r1, r4
   17e3a:	4629      	mov	r1, r5
   17e3c:	bd30      	pop	{r4, r5, pc}
   17e3e:	f094 0f00 	teq	r4, #0
   17e42:	f483 1380 	eor.w	r3, r3, #1048576	; 0x100000
   17e46:	bf06      	itte	eq
   17e48:	f481 1180 	eoreq.w	r1, r1, #1048576	; 0x100000
   17e4c:	3401      	addeq	r4, #1
   17e4e:	3d01      	subne	r5, #1
   17e50:	e74e      	b.n	17cf0 <__adddf3+0x8c>
   17e52:	ea7f 5c64 	mvns.w	ip, r4, asr #21
   17e56:	bf18      	it	ne
   17e58:	ea7f 5c65 	mvnsne.w	ip, r5, asr #21
   17e5c:	d029      	beq.n	17eb2 <__adddf3+0x24e>
   17e5e:	ea94 0f05 	teq	r4, r5
   17e62:	bf08      	it	eq
   17e64:	ea90 0f02 	teqeq	r0, r2
   17e68:	d005      	beq.n	17e76 <__adddf3+0x212>
   17e6a:	ea54 0c00 	orrs.w	ip, r4, r0
   17e6e:	bf04      	itt	eq
   17e70:	4619      	moveq	r1, r3
   17e72:	4610      	moveq	r0, r2
   17e74:	bd30      	pop	{r4, r5, pc}
   17e76:	ea91 0f03 	teq	r1, r3
   17e7a:	bf1e      	ittt	ne
   17e7c:	2100      	movne	r1, #0
   17e7e:	2000      	movne	r0, #0
   17e80:	bd30      	popne	{r4, r5, pc}
   17e82:	ea5f 5c54 	movs.w	ip, r4, lsr #21
   17e86:	d105      	bne.n	17e94 <__adddf3+0x230>
   17e88:	0040      	lsls	r0, r0, #1
   17e8a:	4149      	adcs	r1, r1
   17e8c:	bf28      	it	cs
   17e8e:	f041 4100 	orrcs.w	r1, r1, #2147483648	; 0x80000000
   17e92:	bd30      	pop	{r4, r5, pc}
   17e94:	f514 0480 	adds.w	r4, r4, #4194304	; 0x400000
   17e98:	bf3c      	itt	cc
   17e9a:	f501 1180 	addcc.w	r1, r1, #1048576	; 0x100000
   17e9e:	bd30      	popcc	{r4, r5, pc}
   17ea0:	f001 4500 	and.w	r5, r1, #2147483648	; 0x80000000
   17ea4:	f045 41fe 	orr.w	r1, r5, #2130706432	; 0x7f000000
   17ea8:	f441 0170 	orr.w	r1, r1, #15728640	; 0xf00000
   17eac:	f04f 0000 	mov.w	r0, #0
   17eb0:	bd30      	pop	{r4, r5, pc}
   17eb2:	ea7f 5c64 	mvns.w	ip, r4, asr #21
   17eb6:	bf1a      	itte	ne
   17eb8:	4619      	movne	r1, r3
   17eba:	4610      	movne	r0, r2
   17ebc:	ea7f 5c65 	mvnseq.w	ip, r5, asr #21
   17ec0:	bf1c      	itt	ne
   17ec2:	460b      	movne	r3, r1
   17ec4:	4602      	movne	r2, r0
   17ec6:	ea50 3401 	orrs.w	r4, r0, r1, lsl #12
   17eca:	bf06      	itte	eq
   17ecc:	ea52 3503 	orrseq.w	r5, r2, r3, lsl #12
   17ed0:	ea91 0f03 	teqeq	r1, r3
   17ed4:	f441 2100 	orrne.w	r1, r1, #524288	; 0x80000
   17ed8:	bd30      	pop	{r4, r5, pc}
   17eda:	bf00      	nop

Code Block 4. Arm Software Floating Point Addition Implementation

This isn't even the full code. This is a function that our calculation function has to run each time it wants to add two doubles together. Also, note that it is not just a straight shot of 202 instructions, because you can see that there are loops in the code where ever you see an instruction's mnemonic that starts with the letter b (stands for branch).

Other Use Cases

  • Correlate degrees to radians (assuming degrees are whole numbers)
  • Table of cosine or sine given radians or degrees
    • In the radians case, you will need to create your own trivial hashing function to convert radians to an index
  • Finding a number of bits SET in a 32-bit number
    • Without a lookup table time complexity is O(n) where (n = 32), the number of bits you want to look through
    • With a lookup table, the time complexity is O(1), constant time, and only needs the followin operations
      • 3 bitwise left shifts operations
      • 4 bitwise ANDS operations
      • 4 load from memory addresses
      • 4 binary ADD operations
        • Total of 15 operations total
/* Found this on wikipedia! */

/* Pseudocode of the lookup table 'uint32_t bits_set[256]' */
 /*                    0b00, 0b01, 0b10, 0b11, 0b100, 0b101, ... */
 int bits_set[256] = {    0,    1,    1,    2,     1,     2, // 200+ more entries

 /* (this code assumes that 'int' is an unsigned 32-bits wide integer) */
 int count_ones(unsigned int x) {
    return bits_set[ x        & 255] + bits_set[(x >>  8) & 255]
         + bits_set[(x >> 16) & 255] + bits_set[(x >> 24) & 255];
 }

Code Block 5. Bits set in a 32-bit number (Found this on wikipedia (look up tables))

There are far more use cases then this, but these are a few. 

Lookup Table Decision Tree

Lookup tables can be used as elegant ways to structure information. In this case, they may not provide a speed up but they will associate indexes with something greater, making your code more readable and easier to maintain. In this example, we will be looking at a matrix of function pointers.

Example: Replace Decision Tree

See the function below:

void makeADecisionRobot(bool power_system_nominal, bool no_obstacles_ahead)
{
   if(power_system_nominal && no_obstacles_ahead)
   {
      moveForward();
   }
   else if(power_system_nominal && !no_obstacles_ahead)
   {
      moveOutOfTheWay();
   }
   else if(!power_system_nominal && no_obstacles_ahead)
   {
      slowDown();
   }
   else
   {
      emergencyStop();
   }
}

Code Block 6. Typical Decision Tree 

void (* decision_matrix)(void)[2][2] =
{
   [1][1] = moveForward
   [1][0] = moveOutOfTheWay,
   [0][1] = slowDown,
   [0][0] = emergencyStop,
};

void makeADecisionRobot(bool power_system_nominal, bool no_obstacles_ahead)
{
   decision_matrix[power_system_nominal][no_obstacles_ahead]();
}

Code Block 7. Lookup Table Decision Tree 

 The interesting thing about the decision tree is that it is also more optimal in that, it takes a few instructions to do the look up from memory, then the address of the procedure [function] is looked up an executed, where the former required multiple read instructions and comparison instructions.

This pattern of lookup table will be most useful to us for the interrupts lab assignment.


Revision #6
Created 8 months ago by Admin
Updated 8 months ago by Admin