def to_Sollya(self):
"""
Convert transfer function into two Sollya polynomials (numerator and denomitator)
The result is cached in self._sollya
Returns
- num, den: (sollya object) numerator and denominator of the transfer function
"""
if not self._sollya:
try:
import sollya
except ImportError:
raise ImportError(
"Sollya and SollyaPython are required (but not installed) !"
)
num = sollya.horner(
sum(
sollya.SollyaObject(x) * sollya._x_**i
for i, x in enumerate(array(self.num)[0, :])))
den = sollya.horner(
sum(
sollya.SollyaObject(x) * sollya._x_**i
for i, x in enumerate(array(self.den)[0, :])))
self._sollya = (num, den)
return self._sollya
def determine_error(self):
sollya.settings.display = sollya.hexadecimal
n_pi = self.precision.round_sollya_object(sollya.pi, sollya.RN)
n_invpi = self.precision.round_sollya_object(1 / sollya.pi, sollya.RN)
poly_expr = str(sollya.horner(self.poly_object.get_sollya_object()))
poly_expr = poly_expr.replace("_x_", "z")
poly_expr = poly_expr.replace("z^0x1p1", "z*z")
config = fptaylor.CHECK_CONFIG.copy()
del config["--abs-error"]
config["--opt"] = "bb-eval"
config["--rel-error-threshold"] = "0.0"
config["--intermediate-opt"] = "false"
config["--uncertainty"] = "false"
def generate_fptaylor(x):
x_low = sollya.inf(x)
x_high = sollya.sup(x)
query = "\n".join([
"Variables", " real z in [{},{}];".format(x_low, x_high),
"Definitions", " retval rnd64= {};".format(poly_expr),
"Expressions", " retval;"
])
rnd_rel_err = None
rnd_abs_err = None
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "true",
"--abs-error": "true"
})
rnd_rel_err = float(
res.result["relative_errors"]["final_total"]["value"])
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
except KeyError:
try:
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except KeyError:
pass
if rnd_abs_err is None:
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "false",
"--abs-error": "true"
})
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.sin(sollya.x), x, sollya.relative,
2**-100)
algo_rel_err = sollya.sup(err_int)
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.sin(sollya.x), x, sollya.absolute,
2**-100)
algo_abs_err = sollya.sup(err_int)
if rnd_rel_err is None or str(algo_rel_err) == "error":
rel_err = float("inf")
else:
rel_err = rnd_rel_err + algo_rel_err
abs_err = rnd_abs_err + algo_abs_err
return rel_err, abs_err
def generate_reduction_fptaylor(x):
# get sign and abs_x, must be the same at endpoints
if sollya.sup(x) <= 0:
sign_x_expr = "-1.0"
abs_x_expr = "-x"
abs_x = -x
elif sollya.inf(x) >= 0:
sign_x_expr = "1.0"
abs_x_expr = "x"
abs_x = x
else:
assert False, "Interval must not straddle 0"
# get k, must be the same at endpoints
unround_k = abs_x * n_invpi
k_low = sollya.floor(sollya.inf(unround_k))
k_high = sollya.floor(sollya.sup(unround_k))
if k_low != k_high:
assert False, "Interval must not straddle multples of pi"
k = int(k_low)
part = k % 2
r_expr = "abs_x - whole"
r = abs_x - k * n_pi
z_expr = "r"
z = r
if part == 1:
flipped_poly_expr = "-poly"
else:
flipped_poly_expr = "poly"
x_low = sollya.inf(x)
x_high = sollya.sup(x)
query = "\n".join([
"Variables", " real x in [{},{}];".format(x_low, x_high),
"Definitions", " abs_x rnd64= {};".format(abs_x_expr),
" whole rnd64= {} * {};".format(k, n_pi),
" r rnd64= abs_x - whole;", " z rnd64= {};".format(z_expr),
" poly rnd64= {};".format(poly_expr),
" flipped_poly rnd64= {};".format(flipped_poly_expr),
" retval rnd64= flipped_poly*{};".format(sign_x_expr),
"Expressions", " retval;"
])
rnd_rel_err = None
rnd_abs_err = None
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "true",
"--abs-error": "true"
})
rnd_rel_err = float(
res.result["relative_errors"]["final_total"]["value"])
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
except KeyError:
try:
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except KeyError:
pass
if rnd_abs_err is None:
try:
res = fptaylor.Result(query, {
**config, "--rel-error": "false",
"--abs-error": "true"
})
rnd_abs_err = float(
res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.sin(sollya.x), z, sollya.relative,
2**-100)
algo_rel_err = sollya.sup(err_int)
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.sin(sollya.x), z, sollya.absolute,
2**-100)
algo_abs_err = sollya.sup(err_int)
if rnd_rel_err is None or str(algo_rel_err) == "error":
rel_err = float("inf")
else:
rel_err = rnd_rel_err + algo_rel_err
abs_err = rnd_abs_err + algo_abs_err
return rel_err, abs_err
def split_domain(starting_domain, slivers):
in_domains = [starting_domain]
# abs
out_domains = list()
for I in in_domains:
if sollya.inf(I) < 0 and sollya.sup(I) > 0:
out_domains.append(sollya.Interval(sollya.inf(I), 0))
out_domains.append(sollya.Interval(0, sollya.sup(I)))
else:
out_domains.append(I)
in_domains = out_domains
# k
out_domains = list()
while len(in_domains) > 0:
I = in_domains.pop()
#print("in: [{}, {}]".format(float(sollya.inf(I)), float(sollya.sup(I))))
unround_mult = I * n_invpi
mult_low = sollya.floor(sollya.inf(unround_mult))
mult_high = sollya.floor(sollya.sup(unround_mult))
if mult_low == mult_high or (mult_low == -1
and mult_high == 0):
#print(" accepted")
out_domains.append(I)
continue
if sollya.sup(I) <= 0:
divider_low = (mult_low + 1) * n_pi
divider_high = divider_low - divider_low * 2**-53
else:
divider_high = (mult_low + 1) * n_pi
divider_low = divider_high - divider_high * 2**-53
lower_part = sollya.Interval(sollya.inf(I), divider_low)
upper_part = sollya.Interval(divider_high, sollya.sup(I))
#print(" -> [{}, {}]".format(float(sollya.inf(lower_part)), float(sollya.sup(lower_part))))
#print(" -> [{}, {}]".format(float(sollya.inf(upper_part)), float(sollya.sup(upper_part))))
in_domains.append(lower_part)
in_domains.append(upper_part)
in_domains = out_domains
# subdivide each section into 2**subd sections
for _ in range(slivers):
out_domains = list()
for I in in_domains:
mid = sollya.mid(I)
out_domains.append(sollya.Interval(sollya.inf(I), mid))
out_domains.append(sollya.Interval(mid, sollya.sup(I)))
in_domains = out_domains
in_domains = set(in_domains)
in_domains = sorted(in_domains, key=lambda x: float(sollya.inf(x)))
in_domains = [
d for d in in_domains if sollya.inf(d) != sollya.sup(d)
]
return in_domains
if self.skip_reduction:
starting_domain = sollya.Interval(-n_pi - 2**-7, n_pi + 2**-7)
else:
reduction_k = 20
starting_domain = sollya.Interval(-reduction_k * n_pi,
reduction_k * n_pi)
# analyse each piece
in_domains = split_domain(starting_domain, self.slivers)
errors = list()
for I in in_domains:
if self.skip_reduction:
rel_err, abs_err = generate_fptaylor(I)
else:
rel_err, abs_err = generate_reduction_fptaylor(I)
print("{}\t{}\t{}\t{}".format(float(sollya.inf(I)),
float(sollya.sup(I)), float(abs_err),
float(rel_err)))
errors.append((I, abs_err, rel_err))
def generate_json(errors, domain):
errors = [err for err in errors if err[0] in domain]
errors.sort(key=lambda err: err[2])
epsilon = errors[0][2]
delta = max(err[1] for err in errors)
d = {
"cname": self.function_name,
"delta": float(delta),
"domain": [
float(sollya.inf(domain)),
float(sollya.sup(domain)),
],
"epsilon": float(epsilon),
"operation": "sin"
}
return d
if self.skip_reduction:
d = generate_json(errors,
sollya.Interval(-n_pi - 2**-7, n_pi + 2**-7))
json_str = json.dumps(d, sort_keys=True, indent=4)
json_str = "spec: " + json_str.replace("\n", "\nspec: ")
print(json_str)
else:
specs = list()
for k in range(1, reduction_k):
d = generate_json(errors, sollya.Interval(-k * n_pi, k * n_pi))
specs.append(d)
for i in range(len(specs)):
d = specs[i]
if i == len(specs) - 1:
json_str = json.dumps(d, sort_keys=True, indent=4)
json_str = "spec: " + json_str.replace("\n", "\nspec: ")
print(json_str)
break
nd = specs[i + 1]
if d["epsilon"] == nd["epsilon"] and d["delta"] == nd["delta"]:
continue
json_str = json.dumps(d, sort_keys=True, indent=4)
json_str = "spec: " + json_str.replace("\n", "\nspec: ")
print(json_str)
def determine_error(self):
sollya.settings.display = sollya.hexadecimal
n_log2 = self.precision.round_sollya_object(sollya.log(2), sollya.RN)
poly_expr = str(sollya.horner(self.poly_object.get_sollya_object()))
poly_expr = poly_expr.replace("_x_", "z")
poly_expr = poly_expr.replace("z^0x1p1", "z*z")
config = fptaylor.CHECK_CONFIG.copy()
del config["--abs-error"]
config["--opt"] = "gelpia"
config["--rel-error-threshold"] = "0.0"
config["--intermediate-opt"] = "false"
config["--uncertainty"] = "false"
config["--opt-timeout"] = 100000 # log is returning inf
config["--opt-f-rel-tol"] = "1e0"
config["--opt-f-abs-tol"] = "0.0"
config["--opt-x-rel-tol"] = "0.0"
config["--opt-x-abs-tol"] = "0.0"
def generate_fptaylor(x):
x_low = sollya.inf(x)
x_high = sollya.sup(x)
query = "\n".join(
["Variables",
" real z in [{},{}];".format(x_low, x_high),
"Definitions",
" retval rnd64= {};".format(poly_expr),
"Expressions",
" retval;"])
rnd_rel_err = None
rnd_abs_err = None
try:
res = fptaylor.Result(query, {**config,
"--rel-error": "true",
"--abs-error": "true"})
rnd_rel_err = float(res.result["relative_errors"]["final_total"]["value"])
rnd_abs_err = float(res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
except KeyError:
try:
rnd_abs_err = float(res.result["absolute_errors"]["final_total"]["value"])
except KeyError:
pass
if rnd_abs_err is None:
try:
res = fptaylor.Result(query, {**config,
"--rel-error": "false",
"--abs-error": "true"})
rnd_abs_err = float(res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.log(sollya.x),
x,
sollya.relative,
2**-100)
algo_rel_err = sollya.sup(err_int)
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.log(sollya.x),
x,
sollya.absolute,
2**-100)
algo_abs_err = sollya.sup(err_int)
if rnd_rel_err is None or str(algo_rel_err) == "error":
rel_err = float("inf")
else:
rel_err = rnd_rel_err + algo_rel_err
abs_err = rnd_abs_err + algo_abs_err
return rel_err, abs_err
def generate_reduction_fptaylor(x):
unround_e = sollya.log2(I)
e_low = sollya.floor(sollya.inf(unround_e))
e_high = sollya.floor(sollya.sup(unround_e))
if e_low != e_high:
assert False, "Interval must not stradle a binade"
e = int(e_low) + 1
z = x / (2**e) * 0.5
query = "\n".join(
["Variables",
" real z in [{},{}];".format(sollya.inf(z), sollya.sup(z)),
"Definitions",
" poly rnd64= {};".format(poly_expr),
" retval rnd64= {}*{} + poly;".format(e, n_log2),
"Expressions",
" retval;"])
rnd_rel_err = None
rnd_abs_err = None
try:
res = fptaylor.Result(query, {**config,
"--rel-error": "true",
"--abs-error": "true"})
rnd_rel_err = float(res.result["relative_errors"]["final_total"]["value"])
rnd_abs_err = float(res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
except KeyError:
try:
rnd_abs_err = float(res.result["absolute_errors"]["final_total"]["value"])
except KeyError:
pass
if rnd_abs_err is None:
try:
res = fptaylor.Result(query, {**config,
"--rel-error": "false",
"--abs-error": "true"})
rnd_abs_err = float(res.result["absolute_errors"]["final_total"]["value"])
except AssertionError:
pass
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.log(sollya.x),
z,
sollya.relative,
2**-100)
algo_rel_err = sollya.sup(err_int)
err_int = sollya.supnorm(self.poly_object.get_sollya_object(),
sollya.log(sollya.x),
z,
sollya.absolute,
2**-100)
algo_abs_err = sollya.sup(err_int)
if rnd_rel_err is None or str(algo_rel_err) == "error":
rel_err = float("inf")
else:
rel_err = rnd_rel_err + algo_rel_err
abs_err = rnd_abs_err + algo_abs_err
return rel_err, abs_err
def split_domain(starting_domain, slivers):
in_domains = [starting_domain]
out_domains = list()
while len(in_domains) > 0:
I = in_domains.pop()
unround_e = sollya.log2(I)
e_low = sollya.floor(sollya.inf(unround_e))
e_high = sollya.floor(sollya.sup(unround_e))
#print("in: [{}, {}] ({}, {})".format(float(sollya.inf(I)), float(sollya.sup(I)), int(e_low), int(e_high)))
if e_low == e_high:
#print(" accepted")
out_domains.append(I)
continue
e_range = sollya.Interval(e_low, e_low+1)
I_range = 2**e_range
for _ in range(100):
mid = sollya.mid(I_range)
e = sollya.floor(sollya.log2(mid))
if e == e_low:
I_range = sollya.Interval(mid, sollya.sup(I_range))
else:
I_range = sollya.Interval(sollya.inf(I_range), mid)
divider_high = sollya.sup(I_range)
divider_low = sollya.inf(I_range)
lower_part = sollya.Interval(sollya.inf(I), divider_low)
upper_part = sollya.Interval(divider_high, sollya.sup(I))
#print(" -> [{}, {}]".format(float(sollya.inf(lower_part)), float(sollya.sup(lower_part))))
#print(" -> [{}, {}]".format(float(sollya.inf(upper_part)), float(sollya.sup(upper_part))))
in_domains.append(upper_part)
in_domains.append(lower_part)
in_domains = out_domains
# subdivide each section into 2**subd sections
for _ in range(slivers):
out_domains = list()
for I in in_domains:
mid = sollya.mid(I)
out_domains.append(sollya.Interval(sollya.inf(I), mid))
out_domains.append(sollya.Interval(mid, sollya.sup(I)))
in_domains = out_domains
in_domains = set(in_domains)
in_domains = sorted(in_domains, key=lambda x:float(sollya.inf(x)))
in_domains = [d for d in in_domains if sollya.inf(d) != sollya.sup(d)]
return in_domains
if self.skip_reduction:
starting_domain = sollya.Interval(0.5, 1.0)
else:
reduction_e = 12
starting_domain = sollya.Interval(2**(-reduction_e), 2**reduction_e)
# analyse each piece
in_domains = split_domain(starting_domain, self.slivers)
errors = list()
for I in in_domains:
if self.skip_reduction:
rel_err, abs_err = generate_fptaylor(I)
else:
rel_err, abs_err = generate_reduction_fptaylor(I)
print("{}\t{}\t{}\t{}".format(float(sollya.inf(I)),
float(sollya.sup(I)),
float(abs_err),
float(rel_err)))
errors.append((I, abs_err, rel_err))
def generate_json(errors, domain):
errors = [err for err in errors if err[0] in domain]
errors.sort(key=lambda err: err[2])
epsilon = errors[0][2]
delta = max(err[1] for err in errors)
d = {
"cname": self.function_name,
"delta": float(delta),
"domain": [float(sollya.inf(domain)),
float(sollya.sup(domain)),],
"epsilon": float(epsilon),
"operation": "log"
}
return d
if self.skip_reduction:
d = generate_json(errors, sollya.Interval(0.5, 1.0))
json_str = json.dumps(d, sort_keys=True, indent=4)
json_str = "spec: " + json_str.replace("\n", "\nspec: ")
print(json_str)
else:
specs = list()
for e in range(1, reduction_e):
d = generate_json(errors, sollya.Interval(2**(-e), 2**e))
specs.append(d)
for i in range(len(specs)):
d = specs[i]
if i == len(specs)-1:
json_str = json.dumps(d, sort_keys=True, indent=4)
json_str = "spec: " + json_str.replace("\n", "\nspec: ")
print(json_str)
break
nd = specs[i+1]
if d["epsilon"] == nd["epsilon"] and d["delta"] == nd["delta"]:
continue
json_str = json.dumps(d, sort_keys=True, indent=4)
json_str = "spec: " + json_str.replace("\n", "\nspec: ")
print(json_str)