AoC Day 2 (Gift Shop) in (Haskell &) HardCaml

This problem asks us to compute the sum of invalid product IDs across a set of numeric ranges.

An invalid product ID is a number formed by duplicating a sequence of digits, for example:

• 11, 22
• 1212, 998998
• 824824, 21212121

In general, these numbers all have the form: $\text{dup}(a) = a \times (10^k + 1)$, where a is a k-digit number. Think of this as “write a twice and glue the copies together.”

Reformulation

A straightforward solution would scan every number in each range [lo, hi] and check whether it’s a duplicated pattern. However, that would not be very efficient. We observe that for a fixed k $\text{dup}(a) = a \times (10^k + 1)$ is strictly increasing in a. That means all invalid IDs of a given digit width inside [lo, hi] come from a single, contiguous range of a values.

Starting from: $lo \le a(10^k + 1) \le hi$, we can invert the bounds to get:

loA = max (10^(k-1)) (ceilDiv lo (10^k + 1))
hiA = min (10^k - 1) (hi `div` (10^k + 1))

If loA ≤ hiA, then all invalid IDs for this k are just: $(10^k + 1) \times \sum_{a=loA}^{hiA} a$, which we can compute directly using the arithmetic series formula. No per-ID checking required.

Haskell implementation

Parsing and normalization

parseInput :: String -> [Range]
parseInput =
  pairUp
  . mapMaybe (readMaybe :: String -> Maybe Integer)
  . split ",-"

The input is split on commas and hyphens, parsed into integers, and paired up into (lo, hi) ranges. We then sort and merge overlapping ranges with mergeRanges so we don’t accidentally double-count anything.

Computing the sum

invalidSumInRange :: Range -> Integer
invalidSumInRange (lo, hi) =
  sum [sumForK k | k <- [1 .. maxK]]

For each digit width k, we:

• Compute m = 10^k + 1
• Work out the valid range of a values using division
• Use a closed-form sum to add them up
• Multiply by m to get the contribution to the final answer

This approach avoids iterating over each individual product ID and is fast even for large ranges.

Putting it together

solve :: String -> Integer
solve =
  sum
  . map invalidSumInRange
  . mergeRanges
  . parseInput

Hardware design in HardCaml

Assumptions

Doing string parsing and dynamic lists in hardware is painful and not very interesting here, so we make a few simplifying assumptions:

• Input ranges are already parsed and de-duplicated
• Ranges are streamed in one at a time
• The circuit keeps a running accumulator for the total sum

This lets us focus on the core numeric logic.

Streaming interface

module I = struct
  type 'a t =
    { clock : 'a
    ; clear : 'a
    ; start : 'a
    ; finish : 'a
    ; data_lo : 'a [@bits 64]
    ; data_hi : 'a [@bits 64]
    ; data_in_valid : 'a
    }
end

• start together with data_in_valid loads a new (lo, hi) range
• finish signals that no more ranges are coming
• The output is a With_valid sum that becomes valid when all ranges are processed

State machine

type t =
  | Idle
  | Running
  | Done

Idle

The circuit waits for a valid input range. When one arrives, it initializes:

• k = 1
• p10 = 10
• p10_prev = 1
• the accumulator (which holds the global sum across all ranges)

and transitions to Running.

Running

For each digit width k, the circuit effectively enumerates all k-digit values of a.

It computes:

m = p10 + 1
prod = a * m

In software we’d just divide to find the valid bounds for a, but division is expensive in hardware and not directly available in HardCaml’s combinational logic. Instead, the design leans into iteration.

• Powers of ten are generated using shifts:

  p10_next = (p10 << 3) + (p10 << 1)

• The product prod = a × (10^k + 1) is built incrementally:

  prod <- prod + m
  a    <- a + 1

Each cycle checks:

• prod < lo → not in range yet, keep going
• prod > hi → we’re done with this k, move to the next
• otherwise → add prod to the accumulator

This trades latency for much simpler and cheaper hardware, which is usually a good deal.

Done

Once all digit widths have been processed:

• sum.valid is asserted
• the circuit waits for finish
• then returns to Idle, ready for the next stream

Verification

We use two streaming tests with waveforms for inspections.

As expected, the hardware results exactly match the reference Haskell implementation.

dune build bin/generate.exe @runtest
File "test/test_range_sum.ml", line 1, characters 0-0:
diff --git a/_build/default/test/test_range_sum.ml b/_build/.sandbox/56537e1829c468b9cdc4791eef637e0b/default/test/test_range_sum.ml.corrected
index e05ff98..e7b5050 100644
--- a/_build/default/test/test_range_sum.ml
+++ b/_build/.sandbox/56537e1829c468b9cdc4791eef637e0b/default/test/test_range_sum.ml.corrected
@@ -112,10 +112,110 @@ let run_with_waves ~name ranges =

 let%expect_test "range_sum waveforms (ranges_1)" =
   run_with_waves ~name:"ranges_1" ranges_1;
-  [%expect {| ("Final streaming sum" (!last_sum 1227775554)) |}]
+  [%expect {|
+    ("Final streaming sum" (!last_sum 1227775554))
+    ranges_1
+    ┌Signals─────────────────────┐┌Waves───────────────────────────────────────────────────────┐
+    │                            ││────────────────┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───│
+    │range_sum$a                 ││ 0              │1  │2  │3  │0  │10 │0  │100│0  │10.│0  │10.│
+    │                            ││────────────────┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───│
+    │                            ││────────────────────┬───┬───────────────────────────────────│
+    │range_sum$acc               ││ 0                  │11 │33                                 │
+    │                            ││────────────────────┴───┴───────────────────────────────────│
+    │                            ││────────────┬───────────────────────────────────────────────│
+    │range_sum$hi                ││ 0          │22                                             │
+    │                            ││────────────┴───────────────────────────────────────────────│
+    │range_sum$i$clear           ││────┐                                                       │
+    │                            ││    └───────────────────────────────────────────────────────│
+    │range_sum$i$clock           ││┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ │
+    │                            ││  └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─│
+    │                            ││────────┬───────────────────────────────────────────────────│
+    │range_sum$i$data_hi         ││ 0      │22                                                 │
+    │                            ││────────┴───────────────────────────────────────────────────│
+    │range_sum$i$data_in_valid   ││        ┌───┐                                               │
+    │                            ││────────┘   └───────────────────────────────────────────────│
+    │                            ││────────┬───────────────────────────────────────────────────│
+    │range_sum$i$data_lo         ││ 0      │11                                                 │
+    │                            ││────────┴───────────────────────────────────────────────────│
+    │range_sum$i$finish          ││                                                            │
+    │                            ││────────────────────────────────────────────────────────────│
+    │range_sum$i$start           ││        ┌───┐                                               │
+    │                            ││────────┘   └───────────────────────────────────────────────│
+    │                            ││────────────┬───────────────┬───────┬───────┬───────┬───────│
+    │range_sum$k                 ││ 0          │1              │2      │3      │4      │5      │
+    │                            ││────────────┴───────────────┴───────┴───────┴───────┴───────│
+    │                            ││────────────┬───────────────────────────────────────────────│
+    │range_sum$lo                ││ 0          │11                                             │
+    │                            ││────────────┴───────────────────────────────────────────────│
+    │range_sum$o$sum$valid       ││                                                            │
+    │                            ││────────────────────────────────────────────────────────────│
+    │                            ││────────────────────┬───┬───────────────────────────────────│
+    │range_sum$o$sum$value       ││ 0                  │11 │33                                 │
+    │                            ││────────────────────┴───┴───────────────────────────────────│
+    │                            ││────────────┬───────────────┬───────┬───────┬───────┬───────│
+    │range_sum$p10               ││ 0          │10             │100    │1000   │10000  │100000 │
+    │                            ││────────────┴───────────────┴───────┴───────┴───────┴───────│
+    │                            ││────────────┬───────────────┬───────┬───────┬───────┬───────│
+    │range_sum$p10_prev          ││ 0          │1              │10     │100    │1000   │10000  │
+    │                            ││────────────┴───────────────┴───────┴───────┴───────┴───────│
+    │                            ││────────────────┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───│
+    │range_sum$prod              ││ 0              │11 │22 │33 │0  │10.│0  │10.│0  │10.│0  │10.│
+    │                            ││────────────────┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───│
+    └────────────────────────────┘└────────────────────────────────────────────────────────────┘
+    |}]
 ;;

 let%expect_test "range_sum waveforms (ranges_2)" =
   run_with_waves ~name:"ranges_2" ranges_2;
-  [%expect {| ("Final streaming sum" (!last_sum <computed>)) |}]
+  [%expect {|
+    ("Final streaming sum" (!last_sum 35367539282))
+    ranges_2
+    ┌Signals─────────────────────┐┌Waves───────────────────────────────────────────────────────┐
+    │                            ││────────────────┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───│
+    │range_sum$a                 ││ 0              │1  │2  │3  │4  │5  │6  │7  │8  │9  │10 │0  │
+    │                            ││────────────────┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───│
+    │                            ││────────────────────────────────────────────────────────────│
+    │range_sum$acc               ││ 0                                                          │
+    │                            ││────────────────────────────────────────────────────────────│
+    │                            ││────────────┬───────────────────────────────────────────────│
+    │range_sum$hi                ││ 0          │35281                                          │
+    │                            ││────────────┴───────────────────────────────────────────────│
+    │range_sum$i$clear           ││────┐                                                       │
+    │                            ││    └───────────────────────────────────────────────────────│
+    │range_sum$i$clock           ││┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ ┌─┐ │
+    │                            ││  └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─┘ └─│
+    │                            ││────────┬───────────────────────────────────────────────────│
+    │range_sum$i$data_hi         ││ 0      │35281                                              │
+    │                            ││────────┴───────────────────────────────────────────────────│
+    │range_sum$i$data_in_valid   ││        ┌───┐                                               │
+    │                            ││────────┘   └───────────────────────────────────────────────│
+    │                            ││────────┬───────────────────────────────────────────────────│
+    │range_sum$i$data_lo         ││ 0      │17330                                              │
+    │                            ││────────┴───────────────────────────────────────────────────│
+    │range_sum$i$finish          ││                                                            │
+    │                            ││────────────────────────────────────────────────────────────│
+    │range_sum$i$start           ││        ┌───┐                                               │
+    │                            ││────────┘   └───────────────────────────────────────────────│
+    │                            ││────────────┬───────────────────────────────────────────┬───│
+    │range_sum$k                 ││ 0          │1                                          │2  │
+    │                            ││────────────┴───────────────────────────────────────────┴───│
+    │                            ││────────────┬───────────────────────────────────────────────│
+    │range_sum$lo                ││ 0          │17330                                          │
+    │                            ││────────────┴───────────────────────────────────────────────│
+    │range_sum$o$sum$valid       ││                                                            │
+    │                            ││────────────────────────────────────────────────────────────│
+    │                            ││────────────────────────────────────────────────────────────│
+    │range_sum$o$sum$value       ││ 0                                                          │
+    │                            ││────────────────────────────────────────────────────────────│
+    │                            ││────────────┬───────────────────────────────────────────┬───│
+    │range_sum$p10               ││ 0          │10                                         │100│
+    │                            ││────────────┴───────────────────────────────────────────┴───│
+    │                            ││────────────┬───────────────────────────────────────────┬───│
+    │range_sum$p10_prev          ││ 0          │1                                          │10 │
+    │                            ││────────────┴───────────────────────────────────────────┴───│
+    │                            ││────────────────┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───│
+    │range_sum$prod              ││ 0              │11 │22 │33 │44 │55 │66 │77 │88 │99 │110│0  │
+    │                            ││────────────────┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───│
+    └────────────────────────────┘└────────────────────────────────────────────────────────────┘
+    |}]
 ;;