Given a sorted array of unique positive integers \$A\$ \$(A_0<A_1<\cdots<A_{n-1})\$, and an integer \$t\$, where \$A_0\le t\le A_{n-1}\$. Output the value \$i\$ such that:
- If \$t \in A\$, \$A_i=t\$.
- If \$t \notin A\$, \$ \left(i-\left\lfloor i\right\rfloor\right)\cdot A_{\left\lceil i\right\rceil} +\left(\left\lceil i\right\rceil - i\right)\cdot A_{\left\lfloor i\right\rfloor} = t \$. In other words, linearly interpolate between the indices of the two elements of \$A\$ that most narrowly bound \$t\$.
In the above formula, \$\left\lfloor i\right\rfloor\$ means rounding \$i\$ down to integer; \$\left\lceil i\right\rceil\$ means rounding \$i\$ up to integer.
You may also choose 1-indexed array, though you need to adjust the formula and the test cases below accordingly.
Rules
- This is code-golf, shortest code in bytes wins.
- Floating point errors in output are allowed, but reasonable floating point precision is required. For testcases listed here, your output should be correct with at least 2 decimal places precision.
- Fraction output is allowed as long as fractions are reduced to their lowest terms.
- It is fine to use any built-ins in your language. But if built-ins trivialize the question, consider submitting a non-trivial one too.
Testcases
[42], 42 -> 0
[24, 42], 24 -> 0
[24, 42], 42 -> 1
[1, 3, 5, 7, 8, 10, 12], 1 -> 0
[1, 3, 5, 7, 8, 10, 12], 7 -> 3
[1, 3, 5, 7, 8, 10, 12], 12 -> 6
[24, 42], 33 -> 0.5
[24, 42], 30 -> 0.3333333333333333
[1, 3, 5, 7, 8, 10, 12], 2 -> 0.5
[1, 3, 5, 7, 8, 10, 12], 9 -> 4.5
[100, 200, 400, 800], 128 -> 0.28
[100, 200, 400, 800], 228 -> 1.14
[100, 200, 400, 800], 428 -> 2.07
[1, 3, 7], 9)? \$\endgroup\$InverseFunction[Interpolation[#,InterpolationOrder->1]][N@#2]&. But it's really long, and also doesn't handle the case where A has only one element. \$\endgroup\$