Charcoal, 28 bytes

Ｗ⌈﹪↔θ¹≦⊗θ≦⌊θ≔Ｅ⊕Ｌθ↨…θκ¹θＩ⌕θ⌊θ

Attempt This Online! Link is to verbose version of code. Explanation:

Ｗ⌈﹪↔θ¹≦⊗θ

Double each value until all of them are integers.

≦⌊θ

Convert them from floats to arbitrary precision integers.

≔Ｅ⊕Ｌθ↨…θκ¹θ

Find the cumulative sum.

Ｉ⌕θ⌊θ

Output the index of the minimum.

Maple, 56 bytes

Digits:=99;f:=s->min[index]([(S:=0),seq((S+=i),i=s)])-1;

Input s is a list. All calculations to 99 digits precision (changing to 5000 makes no difference).

Answers to test cases are: 7,10,5,7,9,10,2,4,10,8.

Earlier answer deliberately doing everything in 64 bit floats:

proc(s)S:=0;min[index]([seq((S+=i),i=s)])end;

Input s is an Array with datatype=float[8], i.e., 8-byte IEEE/754 hardware floats.

The test case starting [-9.85558899e-01, -1.18805947e-18 gives 8 not 9, and the (2nd-to-last) test case starting [-4.75110368e-203, 1.02005497e-284 gives 7 not 9. In both these cases the last 3 or 4 (respectively) entries of the cumulative sum are identical. This is because of the loss-of-significance arising from the precision issue. So both answers are equal minima and by @sind's comment are both acceptable.

To get these results one needs to first set the environment to use the right floats by UseHardwareFloats:=true; so perhaps I should add another 24 bytes.

APL(NARS), 24 chars

{⎕ct←0⋄¯1+↑⍸a=⌊/a←0,+\⍵}

0,\⍵ would be the array of partial sums, from that calculate the min, and return the first base 0 index where it is. For the number proposed on test here it is not enought one number float of 64 bits, but a float of 128 bits (as NARS default here), and make sys variable ⎕ct=0 that I presume build ranges for = numbers if it is set not 0. I say errors could be too, but this is true almost always

some test:

{⎕ct←0⋄¯1+↑⍸a=⌊/a←0,+\⍵} 9.20131294e¯9, ¯1.14642378e¯2,  8.76758258e¯10,  2.24754955e¯12, ¯4.98204294e¯01, ¯8.27729005e¯13, ¯1.61108821e¯20,  4.41858056e¯3, 7.97824928e¯6, ¯9.04718754e¯12v    
7
{⎕ct←0⋄¯1+↑⍸a=⌊/a←0,+\⍵}¯2.48038255e¯12,  6.42890672e¯17,  8.57058084e¯08, ¯3.92178008e¯01, ¯1.12017550e¯20,  8.17973856e¯01, ¯6.85909934e¯17,  6.71793290e¯09, 6.88781521e¯04,  3.60301990e¯03v    
5
{⎕ct←0⋄¯1+↑⍸a=⌊/a←0,+\⍵}1.05270211e¯04, ¯5.64364194e¯08,  2.43262585e¯19, ¯4.36773037e¯09,¯4.82114922e¯01, ¯3.65842457e¯02, ¯1.87536618e¯20,  8.81533003e¯20, 4.08686917e¯02,  1.97236499e¯18v    
7
{⎕ct←0⋄¯1+↑⍸a=⌊/a←0,+\⍵}¯9.85558899e¯01, ¯1.18805947e¯18, ¯6.01832197e¯11,  2.23013439e¯02,¯4.82985910e¯01, ¯4.67513770e¯17, ¯4.07790711e¯02, ¯7.91679702e¯04,3.99267831e¯19, ¯4.57447850e¯19v    
10
{⎕ct←0⋄¯1+↑⍸a=⌊/a←0,+\⍵}¯4.75110368e¯203,  1.02005497e¯284, ¯1.26501095e¯213, 3.05297355e¯261,  2.50315890e¯226, ¯2.92734511e¯215,¯7.73205321e¯209,  1.71728306e¯283, ¯4.01406541e¯295,¯2.40473879e¯239v      
10

Google Sheets, 51 bytes

=let(s,scan(,A:A,lambda(a,c,a+c)),match(min(s),s,))

Google Sheets uses IEEE 754 double-precision with a significand of 15.95 decimal digits and an exponent of 11 bits.