def count__split():
parent = operation_count.Computation()
val = operation_count.Float(1.0 + 0.5**27, parent)
high, low = eft._split(val)
assert high.value == 1.0
assert low.value == 0.5**27
assert parent.count == 4
print(" _split(): {}".format(parent.display))
def count_add_eft():
parent = operation_count.Computation()
val1 = operation_count.Float(1.5, parent)
val2 = operation_count.Float(0.5 + 0.5**52, parent)
sum_, error = eft.add_eft(val1, val2)
assert sum_.value == 2.0
assert error.value == 0.5**52
assert parent.count == 6
print(" add_eft(): {}".format(parent.display))
def count__vec_sum():
print("_vec_sum() (6(|p| - 1)):")
for size_p in range(1, 5 + 1):
parent = operation_count.Computation()
p = [operation_count.Float(1.0, parent)] * size_p
eft._vec_sum(p)
assert p[size_p - 1].value == float(size_p)
assert parent.count == 6 * (size_p - 1)
print(" |p| = {}: {}".format(size_p, parent.display))
def count_horner_basic():
print("horner.basic() (2n):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.basic(x, coeffs)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == 2 * degree
print(" degree {}: {}".format(degree, parent.display))
def count_horner_compensated6():
print("horner.compensated_k(..., 6) (312n + 818, n >= 5):")
for degree in range(5, 9 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.compensated_k(x, coeffs, 6)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == horner_expected_total(6, degree)
assert parent.fma_count == horner_expected_fma(6, degree)
print(" degree {}: {}".format(degree, parent.display))
def count_horner_compensated5():
print("horner.compensated_k(..., 5) (152n + 350, n >= 4):")
for degree in range(4, 8 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.compensated_k(x, coeffs, 5)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == horner_expected_total(5, degree)
assert parent.fma_count == horner_expected_fma(5, degree)
print(" degree {}: {}".format(degree, parent.display))
def count_vs_method_basic():
print("vs_method.basic() (5n + 1, w/o binomial):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(0.25, parent)
coeffs = tuple(
operation_count.Float((-1.0)**k, parent)
for k in range(degree + 1))
p = vs_method.basic(x, coeffs)
assert p.value == 0.5**degree
assert parent.count == 5 * degree + 1
print(" degree {}: {}".format(degree, parent.display))
def count_sum_k():
print("sum_k() ((6K - 5)(|p| - 1)):")
for k in (2, 3, 4, 5):
print(" K = {}".format(k))
for size_p in range(1, 5 + 1):
parent = operation_count.Computation()
p = [operation_count.Float(1.0, parent)] * size_p
total = eft.sum_k(p, k)
assert total.value == float(size_p)
assert parent.count == (6 * k - 5) * (size_p - 1)
print(" |p| = {}: {}".format(size_p, parent.display))
def count_de_casteljau_basic():
print("de_casteljau.basic() ((3n^2 + 3n + 2) / 2 = 3 T_n + 1):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(0.25, parent)
coeffs = tuple(
operation_count.Float((-1.0)**k, parent)
for k in range(degree + 1))
p = de_casteljau.basic(x, coeffs)
assert p.value == 0.5**degree
assert parent.count == 3 * (degree * (degree + 1) // 2) + 1
print(" degree {}: {}".format(degree, parent.display))
def count_de_casteljau_compensated4():
print("de_casteljau.compensated4() (65n^2 + 65n + 9 = 130 T_n + 63):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(0.25, parent)
coeffs = tuple(
operation_count.Float((-1.0)**k, parent)
for k in range(degree + 1))
p = de_casteljau.compensated4(x, coeffs)
assert p.value == 0.5**degree
assert parent.count == de_casteljau_expected_total(4, degree)
assert parent.fma_count == de_casteljau_expected_fma(4, degree)
print(" degree {}: {}".format(degree, parent.display))
def count_horner_compensated3():
print("horner.compensated3() (32n + 46, n >= 2):")
for degree in range(2, 6 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.compensated3(x, coeffs)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == horner_expected_total(3, degree)
assert parent.fma_count == horner_expected_fma(3, degree)
print(" degree {}: {}".format(degree, parent.display))
print("horner.compensated_k(..., 3) (32n + 46, n >= 2):")
for degree in range(2, 6 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.compensated_k(x, coeffs, 3)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == horner_expected_total(3, degree)
assert parent.fma_count == horner_expected_fma(3, degree)
print(" degree {}: {}".format(degree, parent.display))
def count_horner_compensated():
print("horner.compensated() (11n + 1):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.compensated(x, coeffs)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == 11 * degree + 1
print(" degree {}: {}".format(degree, parent.display))
# NOTE: This is **the same** as ``horner.compensated()`` but uses
# a different algorithm.
print("horner.compensated_k(..., 2) (12n + 10):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(2.0, parent)
coeffs = (operation_count.Float(1.0, parent), ) * (degree + 1)
p = horner.compensated_k(x, coeffs, 2)
assert p.value == 2.0**(degree + 1) - 1
assert parent.count == horner_expected_total(2, degree)
assert parent.fma_count == horner_expected_fma(2, degree)
print(" degree {}: {}".format(degree, parent.display))
def count_de_casteljau_compensated5():
msg = ("de_casteljau.compensated5() ((231n^2 + 231n + 20) / 2 "
"= 231 T_n + 106):")
print(msg)
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(0.25, parent)
coeffs = tuple(
operation_count.Float((-1.0)**k, parent)
for k in range(degree + 1))
p = de_casteljau.compensated5(x, coeffs)
assert p.value == 0.5**degree
assert parent.count == de_casteljau_expected_total(5, degree)
assert parent.fma_count == de_casteljau_expected_fma(5, degree)
print(" degree {}: {}".format(degree, parent.display))
def count_multiply_eft():
print("multiply_eft():")
for use_fma in (True, False):
parent = operation_count.Computation()
val1 = operation_count.Float(1.0 + 0.5**40, parent)
val2 = operation_count.Float(1.0 - 0.5**40, parent)
product, error = eft.multiply_eft(val1, val2, use_fma=use_fma)
assert product.value == 1.0
assert error.value == -0.5**80
if use_fma:
description = "with FMA: "
assert parent.count == 2
else:
description = "w / out FMA: "
assert parent.count == 17
print(" {} {}".format(description, parent.display))