# beacon_chain # Copyright (c) 2018-2024 Status Research & Development GmbH # Licensed and distributed under either of # * MIT license (license terms in the root directory or at https://opensource.org/licenses/MIT). # * Apache v2 license (license terms in the root directory or at https://www.apache.org/licenses/LICENSE-2.0). # at your option. This file may not be copied, modified, or distributed except according to those terms. {.push raises: [].} # Uncategorized helper functions from the spec import # Status libraries stew/[byteutils, endians2, objects], chronicles, eth/common/[eth_types, eth_types_rlp], eth/rlp, eth/trie/ordered_trie, # Internal "."/[eth2_merkleization, forks, ssz_codec] # TODO although eth2_merkleization already exports ssz_codec, *sometimes* code # fails to compile if the export is not done here also. Exporting rlp avoids a # generics sandwich where rlp/writer.append() is not seen, by a caller outside # this module via compute_execution_block_hash() called from block_processor. export eth2_merkleization, forks, ssz_codec, rlp, eth_types_rlp.append # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/phase0/weak-subjectivity.md#constants const ETH_TO_GWEI = 1_000_000_000.Gwei func toEther*(gwei: Gwei): Ether = (gwei div ETH_TO_GWEI).Ether func toGwei*(eth: Ether): Gwei = distinctBase(eth) * ETH_TO_GWEI type ExecutionHash256* = eth_types.Hash32 ExecutionTransaction* = eth_types.Transaction ExecutionReceipt* = eth_types.Receipt ExecutionWithdrawal* = eth_types.Withdrawal ExecutionDepositRequest* = eth_types.DepositRequest ExecutionWithdrawalRequest* = eth_types.WithdrawalRequest ExecutionConsolidationRequest* = eth_types.ConsolidationRequest ExecutionBlockHeader* = eth_types.Header FinalityCheckpoints* = object justified*: Checkpoint finalized*: Checkpoint func shortLog*(v: FinalityCheckpoints): auto = ( justified: shortLog(v.justified), finalized: shortLog(v.finalized) ) chronicles.formatIt FinalityCheckpoints: it.shortLog # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.7/specs/phase0/beacon-chain.md#integer_squareroot func integer_squareroot*(n: SomeInteger): SomeInteger = ## Return the largest integer ``x`` such that ``x**2 <= n``. doAssert n >= 0'u64 if n == high(uint64): return 4294967295'u64 var x = n y = (x + 1) div 2 while y < x: x = y y = (x + n div x) div 2 x # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.7/specs/phase0/beacon-chain.md#is_active_validator func is_active_validator*(validator: Validator, epoch: Epoch): bool = ## Check if ``validator`` is active. validator.activation_epoch <= epoch and epoch < validator.exit_epoch func is_exited_validator*(validator: Validator, epoch: Epoch): bool = ## Check if ``validator`` is exited. validator.exit_epoch <= epoch func is_withdrawable_validator*(validator: Validator, epoch: Epoch): bool = epoch >= validator.withdrawable_epoch # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.6/specs/phase0/beacon-chain.md#get_active_validator_indices iterator get_active_validator_indices*(state: ForkyBeaconState, epoch: Epoch): ValidatorIndex = for vidx in state.validators.vindices: if is_active_validator(state.validators[vidx], epoch): yield vidx func get_active_validator_indices*(state: ForkyBeaconState, epoch: Epoch): seq[ValidatorIndex] = ## Return the sequence of active validator indices at ``epoch``. var res = newSeqOfCap[ValidatorIndex](state.validators.len) for vidx in get_active_validator_indices(state, epoch): res.add vidx res func get_active_validator_indices_len*(state: ForkyBeaconState, epoch: Epoch): uint64 = for vidx in state.validators.vindices: if is_active_validator(state.validators.item(vidx), epoch): inc result func get_active_validator_indices_len*( state: ForkedHashedBeaconState; epoch: Epoch): uint64 = withState(state): get_active_validator_indices_len(forkyState.data, epoch) # https://github.com/ethereum/consensus-specs/blob/v1.4.0/specs/phase0/beacon-chain.md#get_current_epoch func get_current_epoch*(state: ForkyBeaconState): Epoch = ## Return the current epoch. state.slot.epoch # https://github.com/ethereum/consensus-specs/blob/v1.4.0/specs/phase0/beacon-chain.md#get_current_epoch func get_current_epoch*(state: ForkedHashedBeaconState): Epoch = ## Return the current epoch. withState(state): get_current_epoch(forkyState.data) # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.6/specs/phase0/beacon-chain.md#get_previous_epoch func get_previous_epoch*( state: ForkyBeaconState | ForkedHashedBeaconState): Epoch = ## Return the previous epoch (unless the current epoch is ``GENESIS_EPOCH``). get_previous_epoch(get_current_epoch(state)) # https://github.com/ethereum/consensus-specs/blob/v1.4.0/specs/phase0/beacon-chain.md#get_randao_mix func get_randao_mix*(state: ForkyBeaconState, epoch: Epoch): Eth2Digest = ## Return the randao mix at a recent ``epoch``. state.randao_mixes[epoch mod EPOCHS_PER_HISTORICAL_VECTOR] func bytes_to_uint32*(data: openArray[byte]): uint32 = doAssert data.len == 4 # Little-endian data representation uint32.fromBytesLE(data) func bytes_to_uint64*(data: openArray[byte]): uint64 = doAssert data.len == 8 # Little-endian data representation uint64.fromBytesLE(data) func uint_to_bytes*(x: uint64): array[8, byte] = toBytesLE(x) func uint_to_bytes*(x: uint32): array[4, byte] = toBytesLE(x) func uint_to_bytes*(x: uint16): array[2, byte] = toBytesLE(x) func uint_to_bytes*(x: uint8): array[1, byte] = toBytesLE(x) # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.7/specs/phase0/beacon-chain.md#compute_domain func compute_domain*( domain_type: DomainType, fork_version: Version, genesis_validators_root: Eth2Digest = ZERO_HASH): Eth2Domain = ## Return the domain for the ``domain_type`` and ``fork_version``. # # TODO toOpenArray can't be used from JavaScript backend # https://github.com/nim-lang/Nim/issues/15952 let fork_data_root = compute_fork_data_root(fork_version, genesis_validators_root) result[0..3] = domain_type.data result[4..31] = fork_data_root.data.toOpenArray(0, 27) # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/phase0/beacon-chain.md#get_domain func get_domain*( fork: Fork, domain_type: DomainType, epoch: Epoch, genesis_validators_root: Eth2Digest): Eth2Domain = ## Return the signature domain (fork version concatenated with domain type) ## of a message. let fork_version = if epoch < fork.epoch: fork.previous_version else: fork.current_version compute_domain(domain_type, fork_version, genesis_validators_root) func get_domain*( state: ForkyBeaconState, domain_type: DomainType, epoch: Epoch): Eth2Domain = ## Return the signature domain (fork version concatenated with domain type) ## of a message. get_domain(state.fork, domain_type, epoch, state.genesis_validators_root) # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.6/specs/phase0/beacon-chain.md#compute_signing_root func compute_signing_root*(ssz_object: auto, domain: Eth2Domain): Eth2Digest = ## Return the signing root for the corresponding signing data. let domain_wrapped_object = SigningData( object_root: hash_tree_root(ssz_object), domain: domain ) hash_tree_root(domain_wrapped_object) # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/phase0/beacon-chain.md#get_seed func get_seed*( state: ForkyBeaconState, epoch: Epoch, domain_type: DomainType, mix: Eth2Digest): Eth2Digest = ## Return the seed at ``epoch``. var seed_input : array[4+8+32, byte] seed_input[0..3] = domain_type.data seed_input[4..11] = uint_to_bytes(epoch.uint64) seed_input[12..43] = mix.data eth2digest(seed_input) func get_seed*(state: ForkyBeaconState, epoch: Epoch, domain_type: DomainType): Eth2Digest = # Detect potential underflow static: doAssert EPOCHS_PER_HISTORICAL_VECTOR > MIN_SEED_LOOKAHEAD let mix = get_randao_mix(state, # Avoid underflow epoch + EPOCHS_PER_HISTORICAL_VECTOR - MIN_SEED_LOOKAHEAD - 1) state.get_seed(epoch, domain_type, mix) # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/altair/beacon-chain.md#add_flag func add_flag*(flags: ParticipationFlags, flag_index: TimelyFlag): ParticipationFlags = let flag = ParticipationFlags(1'u8 shl ord(flag_index)) flags or flag # https://github.com/ethereum/consensus-specs/blob/v1.4.0/specs/altair/beacon-chain.md#has_flag func has_flag*(flags: ParticipationFlags, flag_index: TimelyFlag): bool = let flag = ParticipationFlags(1'u8 shl ord(flag_index)) (flags and flag) == flag # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/deneb/p2p-interface.md#verify_blob_sidecar_inclusion_proof func verify_blob_sidecar_inclusion_proof*( blob_sidecar: BlobSidecar): Result[void, string] = let gindex = kzg_commitment_inclusion_proof_gindex(blob_sidecar.index) if not is_valid_merkle_branch( hash_tree_root(blob_sidecar.kzg_commitment), blob_sidecar.kzg_commitment_inclusion_proof, KZG_COMMITMENT_INCLUSION_PROOF_DEPTH, get_subtree_index(gindex), blob_sidecar.signed_block_header.message.body_root): return err("BlobSidecar: inclusion proof not valid") ok() func create_blob_sidecars*( forkyBlck: deneb.SignedBeaconBlock | electra.SignedBeaconBlock, kzg_proofs: KzgProofs, blobs: Blobs): seq[BlobSidecar] = template kzg_commitments: untyped = forkyBlck.message.body.blob_kzg_commitments doAssert kzg_proofs.len == blobs.len doAssert kzg_proofs.len == kzg_commitments.len var res = newSeqOfCap[BlobSidecar](blobs.len) let signedBlockHeader = forkyBlck.toSignedBeaconBlockHeader() for i in 0 ..< blobs.lenu64: var sidecar = BlobSidecar( index: i, blob: blobs[i], kzg_commitment: kzg_commitments[i], kzg_proof: kzg_proofs[i], signed_block_header: signedBlockHeader) forkyBlck.message.body.build_proof( kzg_commitment_inclusion_proof_gindex(i), sidecar.kzg_commitment_inclusion_proof).expect("Valid gindex") res.add(sidecar) res # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.5/specs/altair/light-client/sync-protocol.md#is_sync_committee_update template is_sync_committee_update*(update: SomeForkyLightClientUpdate): bool = when update is SomeForkyLightClientUpdateWithSyncCommittee: update.next_sync_committee_branch != static(default(typeof(update.next_sync_committee_branch))) else: false # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.5/specs/altair/light-client/sync-protocol.md#is_finality_update template is_finality_update*(update: SomeForkyLightClientUpdate): bool = when update is SomeForkyLightClientUpdateWithFinality: update.finality_branch != static(default(typeof(update.finality_branch))) else: false # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.3/specs/altair/light-client/sync-protocol.md#is_next_sync_committee_known template is_next_sync_committee_known*(store: ForkyLightClientStore): bool = store.next_sync_committee != static(default(typeof(store.next_sync_committee))) # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.5/specs/altair/light-client/sync-protocol.md#get_safety_threshold func get_safety_threshold*(store: ForkyLightClientStore): uint64 = max( store.previous_max_active_participants, store.current_max_active_participants ) div 2 # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.5/specs/altair/light-client/sync-protocol.md#is_better_update type LightClientUpdateMetadata* = object attested_slot*, finalized_slot*, signature_slot*: Slot has_sync_committee*, has_finality*: bool num_active_participants*: uint64 func toMeta*(update: SomeForkyLightClientUpdate): LightClientUpdateMetadata = var meta {.noinit.}: LightClientUpdateMetadata meta.attested_slot = update.attested_header.beacon.slot meta.finalized_slot = when update is SomeForkyLightClientUpdateWithFinality: update.finalized_header.beacon.slot else: GENESIS_SLOT meta.signature_slot = update.signature_slot meta.has_sync_committee = when update is SomeForkyLightClientUpdateWithSyncCommittee: update.is_sync_committee_update else: false meta.has_finality = when update is SomeForkyLightClientUpdateWithFinality: update.is_finality_update else: false meta.num_active_participants = update.sync_aggregate.num_active_participants.uint64 meta template toMeta*( update: SomeForkedLightClientUpdate): LightClientUpdateMetadata = withForkyObject(update): when lcDataFork > LightClientDataFork.None: forkyObject.toMeta() else: default(LightClientUpdateMetadata) func is_better_data*(new_meta, old_meta: LightClientUpdateMetadata): bool = # Compare supermajority (> 2/3) sync committee participation let new_has_supermajority = hasSupermajoritySyncParticipation(new_meta.num_active_participants) old_has_supermajority = hasSupermajoritySyncParticipation(old_meta.num_active_participants) if new_has_supermajority != old_has_supermajority: return new_has_supermajority > old_has_supermajority if not new_has_supermajority: if new_meta.num_active_participants != old_meta.num_active_participants: return new_meta.num_active_participants > old_meta.num_active_participants # Compare presence of relevant sync committee let new_has_relevant_sync_committee = new_meta.has_sync_committee and new_meta.attested_slot.sync_committee_period == new_meta.signature_slot.sync_committee_period old_has_relevant_sync_committee = old_meta.has_sync_committee and old_meta.attested_slot.sync_committee_period == old_meta.signature_slot.sync_committee_period if new_has_relevant_sync_committee != old_has_relevant_sync_committee: return new_has_relevant_sync_committee > old_has_relevant_sync_committee # Compare indication of any finality if new_meta.has_finality != old_meta.has_finality: return new_meta.has_finality > old_meta.has_finality # Compare sync committee finality if new_meta.has_finality: let new_has_sync_committee_finality = new_meta.finalized_slot.sync_committee_period == new_meta.attested_slot.sync_committee_period old_has_sync_committee_finality = old_meta.finalized_slot.sync_committee_period == old_meta.attested_slot.sync_committee_period if new_has_sync_committee_finality != old_has_sync_committee_finality: return new_has_sync_committee_finality > old_has_sync_committee_finality # Tiebreaker 1: Sync committee participation beyond supermajority if new_meta.num_active_participants != old_meta.num_active_participants: return new_meta.num_active_participants > old_meta.num_active_participants # Tiebreaker 2: Prefer older data (fewer changes to best data) new_meta.attested_slot < old_meta.attested_slot template is_better_update*[ A, B: SomeForkyLightClientUpdate | ForkedLightClientUpdate]( new_update: A, old_update: B): bool = is_better_data(toMeta(new_update), toMeta(old_update)) # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.5/specs/altair/light-client/p2p-interface.md#getlightclientbootstrap func contextEpoch*(bootstrap: ForkyLightClientBootstrap): Epoch = bootstrap.header.beacon.slot.epoch # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.5/specs/altair/light-client/p2p-interface.md#lightclientupdatesbyrange # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.5/specs/altair/light-client/p2p-interface.md#getlightclientfinalityupdate # https://github.com/ethereum/consensus-specs/blob/v1.4.0-beta.5/specs/altair/light-client/p2p-interface.md#getlightclientoptimisticupdate func contextEpoch*(update: SomeForkyLightClientUpdate): Epoch = update.attested_header.beacon.slot.epoch # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/bellatrix/beacon-chain.md#is_merge_transition_complete func is_merge_transition_complete*( state: bellatrix.BeaconState | capella.BeaconState | deneb.BeaconState | electra.BeaconState): bool = const defaultExecutionPayloadHeader = default(typeof(state.latest_execution_payload_header)) state.latest_execution_payload_header != defaultExecutionPayloadHeader # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/sync/optimistic.md#helpers func is_execution_block*(blck: SomeForkyBeaconBlock): bool = when typeof(blck).kind >= ConsensusFork.Bellatrix: const defaultExecutionPayload = default(typeof(blck.body.execution_payload)) blck.body.execution_payload != defaultExecutionPayload else: false # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/bellatrix/beacon-chain.md#is_merge_transition_block func is_merge_transition_block( state: bellatrix.BeaconState | capella.BeaconState | deneb.BeaconState | electra.BeaconState, body: bellatrix.BeaconBlockBody | bellatrix.TrustedBeaconBlockBody | bellatrix.SigVerifiedBeaconBlockBody | capella.BeaconBlockBody | capella.TrustedBeaconBlockBody | capella.SigVerifiedBeaconBlockBody | deneb.BeaconBlockBody | deneb.TrustedBeaconBlockBody | deneb.SigVerifiedBeaconBlockBody | electra.BeaconBlockBody | electra.TrustedBeaconBlockBody | electra.SigVerifiedBeaconBlockBody): bool = const defaultExecutionPayload = default(typeof(body.execution_payload)) not is_merge_transition_complete(state) and body.execution_payload != defaultExecutionPayload # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/bellatrix/beacon-chain.md#is_execution_enabled func is_execution_enabled*( state: bellatrix.BeaconState | capella.BeaconState | deneb.BeaconState | electra.BeaconState, body: bellatrix.BeaconBlockBody | bellatrix.TrustedBeaconBlockBody | bellatrix.SigVerifiedBeaconBlockBody | capella.BeaconBlockBody | capella.TrustedBeaconBlockBody | capella.SigVerifiedBeaconBlockBody | deneb.BeaconBlockBody | deneb.TrustedBeaconBlockBody | deneb.SigVerifiedBeaconBlockBody | electra.BeaconBlockBody | electra.TrustedBeaconBlockBody | electra.SigVerifiedBeaconBlockBody): bool = is_merge_transition_block(state, body) or is_merge_transition_complete(state) # https://github.com/ethereum/consensus-specs/blob/v1.5.0-alpha.8/specs/bellatrix/beacon-chain.md#compute_timestamp_at_slot func compute_timestamp_at_slot*(state: ForkyBeaconState, slot: Slot): uint64 = # Note: This function is unsafe with respect to overflows and underflows. let slots_since_genesis = slot - GENESIS_SLOT state.genesis_time + slots_since_genesis * SECONDS_PER_SLOT template append*(w: var RlpWriter, v: bellatrix.Transaction) = w.appendRawBytes(distinctBase v) template append*(w: var RlpWriter, withdrawal: capella.Withdrawal) = w.appendRecordType(ExecutionWithdrawal( index: withdrawal.index, validatorIndex: withdrawal.validator_index, address: EthAddress withdrawal.address.data, amount: distinctBase(withdrawal.amount))) proc computeTransactionsTrieRoot( payload: ForkyExecutionPayload): ExecutionHash256 = orderedTrieRoot(payload.transactions.asSeq) func append*(w: var RlpWriter, request: electra.DepositRequest) = w.append ExecutionDepositRequest( pubkey: Bytes48 request.pubkey.blob, withdrawalCredentials: Bytes32 request.withdrawal_credentials.data, amount: distinctBase(request.amount), signature: Bytes96 request.signature.blob, index: request.index) func append*(w: var RlpWriter, request: electra.WithdrawalRequest) = w.append ExecutionWithdrawalRequest( sourceAddress: Address request.source_address.data, validatorPubkey: Bytes48 request.validator_pubkey.blob, amount: distinctBase(request.amount)) func append*(w: var RlpWriter, request: electra.ConsolidationRequest) = w.append ExecutionConsolidationRequest( sourceAddress: Address request.source_address.data, sourcePubkey: Bytes48 request.source_pubkey.blob, targetPubkey: Bytes48 request.target_pubkey.blob) # https://eips.ethereum.org/EIPS/eip-7685 proc computeRequestsTrieRoot( requests: electra.ExecutionRequests): ExecutionHash256 = let n = requests.deposits.len + requests.withdrawals.len + requests.consolidations.len var b = OrderedTrieRootBuilder.init(n) static: doAssert DEPOSIT_REQUEST_TYPE < WITHDRAWAL_REQUEST_TYPE doAssert WITHDRAWAL_REQUEST_TYPE < CONSOLIDATION_REQUEST_TYPE b.add(requests.deposits.asSeq) # EIP-6110 b.add(requests.withdrawals.asSeq) # EIP-7002 b.add(requests.consolidations.asSeq) # EIP-7251 b.rootHash() proc blockToBlockHeader*(blck: ForkyBeaconBlock): ExecutionBlockHeader = template payload: auto = blck.body.execution_payload static: # `GasInt` is signed. We only use it for hashing. doAssert sizeof(GasInt) == sizeof(payload.gas_limit) doAssert sizeof(GasInt) == sizeof(payload.gas_used) let txRoot = payload.computeTransactionsTrieRoot() withdrawalsRoot = when typeof(payload).kind >= ConsensusFork.Capella: Opt.some orderedTrieRoot(payload.withdrawals.asSeq) else: Opt.none(ExecutionHash256) blobGasUsed = when typeof(payload).kind >= ConsensusFork.Deneb: Opt.some payload.blob_gas_used else: Opt.none(uint64) excessBlobGas = when typeof(payload).kind >= ConsensusFork.Deneb: Opt.some payload.excess_blob_gas else: Opt.none(uint64) parentBeaconBlockRoot = when typeof(payload).kind >= ConsensusFork.Deneb: Opt.some ExecutionHash256(blck.parent_root.data) else: Opt.none(ExecutionHash256) requestsRoot = when typeof(payload).kind >= ConsensusFork.Electra: Opt.some blck.body.execution_requests.computeRequestsTrieRoot() else: Opt.none(ExecutionHash256) ExecutionBlockHeader( parentHash : payload.parent_hash.to(Hash32), ommersHash : EMPTY_UNCLE_HASH, coinbase : EthAddress payload.fee_recipient.data, stateRoot : payload.state_root.to(Root), transactionsRoot : txRoot, receiptsRoot : payload.receipts_root.to(Root), logsBloom : Bloom payload.logs_bloom.data.to(Bloom), difficulty : default(DifficultyInt), number : payload.block_number, gasLimit : payload.gas_limit, gasUsed : payload.gas_used, timestamp : EthTime(payload.timestamp), extraData : payload.extra_data.asSeq, mixHash : Bytes32 payload.prev_randao.data, # EIP-4399 `mixHash` -> `prevRandao` nonce : default(Bytes8), baseFeePerGas : Opt.some payload.base_fee_per_gas, withdrawalsRoot : withdrawalsRoot, blobGasUsed : blobGasUsed, # EIP-4844 excessBlobGas : excessBlobGas, # EIP-4844 parentBeaconBlockRoot : parentBeaconBlockRoot, # EIP-4788 requestsRoot : requestsRoot) # EIP-7685 proc compute_execution_block_hash*(blck: ForkyBeaconBlock): Eth2Digest = rlpHash(blockToBlockHeader(blck)).to(Eth2Digest) from std/math import exp, ln from std/sequtils import foldl func ln_binomial(n, k: int): float64 = if k > n: low(float64) else: template ln_factorial(n: int): float64 = (2 .. n).foldl(a + ln(b.float64), 0.0) ln_factorial(n) - ln_factorial(k) - ln_factorial(n - k) func hypergeom_cdf*(k: int, population: int, successes: int, draws: int): float64 = if k < draws + successes - population: 0.0 elif k >= min(successes, draws): 1.0 else: let ln_denom = ln_binomial(population, draws) (0 .. k).foldl(a + exp( ln_binomial(successes, b) + ln_binomial(population - successes, draws - b) - ln_denom), 0.0)