exponentiation.precision_ledger module

Module for storing and computing the required number of terms in Taylor approximations of the exponentiation function in order to bound the absolute error.

class exponentiation.precision_ledger.PrecisionLedger

Bases: object

Object for computing and storing the required number of terms in the Taylor approximation of an exponentiation to achieve a certain precision.

__init__()

Initialize instance of PrecisionLedger.

nr_terms(base, bit_precision, nr_terms_cap=15)

Returns the number of terms in the Taylor approximation of an exponentiation to achieve a certain precision.

More precisely, returns the number n required to approximate base**z by the finite sum of ln(base)**i / i! for i from 0 to n where the absolute error is bounded.

Parameters:

• base (int) – Base of the exponentiation.

• bit_precision (int) – Require an 2 ** -bit_precision upper bound on the absolute error of the approximation for all exponents with modulus at most one.

• nr_terms_cap (int) – Upper bound on the number of terms that is returned.

Return type:

int

Returns:

Minimum number of terms in the Taylor approximation to satisfy accuracy constraints.

class exponentiation.precision_ledger.PrecisionLedgerIndex(base, bit_precision, nr_terms_cap)

Bases: object

Multi-key index for entries in the PrecisionLedger.

base: int

bit_precision: int

nr_terms_cap: int

class exponentiation.precision_ledger.PrecisionLedgerResult(nr_terms)

Bases: object

Stores computational results for the PrecisionLedger.

nr_terms: int

exception exponentiation.precision_ledger.PrecisionNrTermsException

Bases: Exception

Required precision was not achieved with given maximum amount of terms.

exponentiation.precision_ledger.calculate_max_sum_index(base, bit_precision, nr_terms_cap)

Compute the number of terms of the Taylor approximation of base ** z so that the absolute error is at most 2 ** -self.bit_precision.

Guarantees hold for $|z|<1$.

Parameters:

• base (int) – Base of the exponentiation.

• nr_terms_cap (int) – Upper bound on the number of terms that is returned.

Return type:

int

Returns:

Minimal number of terms in the Taylor approximation to achieve the desired precision.

Raises:

PrecisionNrTermsException – Raised when required precision cannot be reached