_sqrt.prg

#
# Finds the rounded down square root of 8 bit value value in 14.
# Uses self-modifying code to store result, so
# instruction 11 (LDI 0) and the two values (14, 15) needs to be reset between runs.
#

LDA 14
SUB 15
JPC 11
STA 14
LDI 1
ADD 11
STA 11
LDI 2
ADD 15
STA 15
JMP 0
LDI 0
OUT 0
HLT 0
14: 143
15: 1

_sqrt.prg.bin

00011110
00111111
01111011
01001110
01010001
00101011
01001011
01010010
00101111
01001111
01100000
01010000
11100000
11110000
1110: 10001111
1111: 00000001

sqrt.go

package main

import (
	"fmt"
	"math"
)

func main() {
	for i := 0; i < 256; i++ {
		ref := math.Sqrt(float64(i))
		fmt.Println(i, sqrt(uint8(i)), uint8(ref))
	}
}

func sqrt(x uint8) uint8 {
	r1 := x        // input
	r2 := uint8(1) // number to subtract
	r3 := uint8(0) // result
	for {
		// We don't have JC, so this will have to do.
		if r2 > r1 {
			return r3
		}
		r1 = r1 - r2
		r2 += 2
		r3++
	}
	return 0
}

https://play.golang.org/p/k2o8_U_OpSj