Here is how GPT-$ Turbo performs long multiplication:
multiply 12.23341 by -123.23354 using long multiplication method.
To multiply 12.23341 by -123.23354 using the long multiplication method, we first ignore the decimal points and treat the numbers as integers. We'll reintroduce the decimal point at the end. Here's the multiplication step by step:
```
1223341 (this is 12.23341 with the decimal removed)
× -12323354 (this is -123.23354 with the decimal removed)
--------------
1223341 (multiplied by 4)
0000000 (multiplied by 5, shifted one position to the left)
1223341 (multiplied by 3, shifted two positions to the left)
3660023 (multiplied by 3, shifted three positions to the left)
4886682 (multiplied by 2, shifted four positions to the left)
0000000 (multiplied by 2, shifted five positions to the left)
1223341 (multiplied by 1, shifted six positions to the left)
1223341 (multiplied by 1, shifted seven positions to the left)
0000000 (multiplied by 2, shifted eight positions to the left)
0000000 (multiplied by 3, shifted nine positions to the left)
--------------
15077944092654 (sum of all the above products)
```
Now, we need to place the decimal point. The original numbers had 5 and 8 decimal places, respectively, so the result should have 5+8 = 13 decimal places.
So, 15077944092654 becomes 1507.7944092654.
Since one of the original numbers was negative, the result is also negative:
-1507.7944092654
Therefore, 12.23341 multiplied by -123.23354 equals -1507.7944092654.
Compare with correct result: -1507.5664205714
Here is how I experimented with long multiplication on GPT-4 a month ago (5 shot, GPT-4 made mistakes on addition in 3rd some time 4th digit, regardless of decimal point position):
Multiply -0.9923 by -0.9923 using long multiplication.Solution: multiply individual digits in reverse order, increasing the order of magnitude of the result for each next digit in the first number : 3310*(0+0)=9, 3210*(0+1)=60, 3910*(0+2)=2700, 3910*(0+3)=27000, 3010*(0+4)=0, step's subtotal[0]=29769; 2310*(1+0)=60, 2210*(1+1)=400, 2910*(1+2)=18000, 2910*(1+3)=180000, 2010*(1+4)=0, step's subtotal[1]=198460; 9310*(2+0)=2700, 9210*(2+1)=18000, 9910*(2+2)=810000, 9910*(2+3)=8100000, 9010*(2+4)=0, step's subtotal[2]=8930700; 9310*(3+0)=27000, 9210*(3+1)=180000, 9910*(3+2)=8100000, 9910*(3+3)=81000000, 9010*(3+4)=0, step's subtotal[3]=89307000; 0310*(4+0)=0, 0210*(4+1)=0, 0910*(4+2)=0, 0910*(4+3)=0, 0010*(4+4)=0, step's subtotal[4]=0; Sum of partial results: 29769+198460+8930700+89307000+0 = 98465929. Set the decimal point position in the result by adding the decimal places of both numbers (4+4=8), counting from the right. Final result: -0.9923*-0.9923=0.98465929
I was able to tune the LLaMA 2 with QLoRA to produce viable results only with precision up to 4th digit after decimal point, however difference in length of mantissa cases wrong result.
Here is how GPT-4 performs long multiplication: https://www.phind.com/search?q=multiply%2012.23341%20by%20-1...
Here is how GPT-$ Turbo performs long multiplication: multiply 12.23341 by -123.23354 using long multiplication method. To multiply 12.23341 by -123.23354 using the long multiplication method, we first ignore the decimal points and treat the numbers as integers. We'll reintroduce the decimal point at the end. Here's the multiplication step by step: ``` 1223341 (this is 12.23341 with the decimal removed) × -12323354 (this is -123.23354 with the decimal removed) -------------- 1223341 (multiplied by 4) 0000000 (multiplied by 5, shifted one position to the left) 1223341 (multiplied by 3, shifted two positions to the left) 3660023 (multiplied by 3, shifted three positions to the left) 4886682 (multiplied by 2, shifted four positions to the left) 0000000 (multiplied by 2, shifted five positions to the left) 1223341 (multiplied by 1, shifted six positions to the left) 1223341 (multiplied by 1, shifted seven positions to the left) 0000000 (multiplied by 2, shifted eight positions to the left) 0000000 (multiplied by 3, shifted nine positions to the left) -------------- 15077944092654 (sum of all the above products) ``` Now, we need to place the decimal point. The original numbers had 5 and 8 decimal places, respectively, so the result should have 5+8 = 13 decimal places. So, 15077944092654 becomes 1507.7944092654. Since one of the original numbers was negative, the result is also negative: -1507.7944092654
Therefore, 12.23341 multiplied by -123.23354 equals -1507.7944092654.
Compare with correct result: -1507.5664205714
Here is how I experimented with long multiplication on GPT-4 a month ago (5 shot, GPT-4 made mistakes on addition in 3rd some time 4th digit, regardless of decimal point position):
Multiply -0.9923 by -0.9923 using long multiplication.Solution: multiply individual digits in reverse order, increasing the order of magnitude of the result for each next digit in the first number : 3310*(0+0)=9, 3210*(0+1)=60, 3910*(0+2)=2700, 3910*(0+3)=27000, 3010*(0+4)=0, step's subtotal[0]=29769; 2310*(1+0)=60, 2210*(1+1)=400, 2910*(1+2)=18000, 2910*(1+3)=180000, 2010*(1+4)=0, step's subtotal[1]=198460; 9310*(2+0)=2700, 9210*(2+1)=18000, 9910*(2+2)=810000, 9910*(2+3)=8100000, 9010*(2+4)=0, step's subtotal[2]=8930700; 9310*(3+0)=27000, 9210*(3+1)=180000, 9910*(3+2)=8100000, 9910*(3+3)=81000000, 9010*(3+4)=0, step's subtotal[3]=89307000; 0310*(4+0)=0, 0210*(4+1)=0, 0910*(4+2)=0, 0910*(4+3)=0, 0010*(4+4)=0, step's subtotal[4]=0; Sum of partial results: 29769+198460+8930700+89307000+0 = 98465929. Set the decimal point position in the result by adding the decimal places of both numbers (4+4=8), counting from the right. Final result: -0.9923*-0.9923=0.98465929
I was able to tune the LLaMA 2 with QLoRA to produce viable results only with precision up to 4th digit after decimal point, however difference in length of mantissa cases wrong result.