double_interval.h
#include <flint/double_interval.h>This module implements interval arithmetic with double endpoints, for computing cheap but rigorous enclosures where the precision of a double is enough, such as bounds on the magnitude of an integrand or of the terms of a series. It is used by the numerical integration code of arb_hypgeom (for example arb_hypgeom_1f1_integration). Unlike the IEEE 1788 approach, which switches the rounding mode of the processor, the operations here compute the endpoints in the default round-to-nearest mode and then widen them by a few ulps, which is valid whether or not the operation was exact. The method and its correctness are explained in Interval arithmetic with doubles.
For rigorous arithmetic at higher precision, use the arb_t balls, or the double-word to quad-double balls d2b_t, ..., d4b_t of dfloat.
Types#
A struct with two double members a and b, representing the extended real interval [a, b] (the endpoints may be infinite). Unless stated otherwise, functions require a \le b and that neither endpoint is NaN. The type is passed and returned by value.
DI_CHECK(x)Throws an exception unless x.a <= x.b (in particular, if an endpoint is NaN).
Construction and conversion#
di_t di_interval(double a, double b)Returns the interval [a, b]. Throws an exception unless a \le b (in particular if a or b is NaN).
di_t arb_get_di(const arb_t x)Returns an interval containing the ball x: its lower and upper endpoints, rounded down and up to doubles. A ball with a NaN midpoint gives [-\infty, +\infty], and an infinite radius gives infinite endpoints.
void arb_set_di(arb_t res, di_t x, slong prec)Sets \mathit{res} to a ball containing the interval x, with the midpoint rounded to \mathit{prec} bits (by arb_set_interval_arf).
Prints x to standard output as [a, b] with 17 significant digits per endpoint. The decimal endpoints are not rounded outward, so the printed interval need not contain x.
double d_randtest2(flint_rand_t state)Returns a random non-NaN double \pm m 2^e with m from d_randtest and e uniform in [-1200, 1199], so that the result can also be zero (after underflow), subnormal or infinite.
di_t di_randtest(flint_rand_t state)Returns an interval with two random endpoints from d_randtest2, ordered.
Arithmetic#
The di_fast_ functions compute the endpoints with the default rounding (assumed to be to nearest) and then move the lower endpoint down and the upper endpoint up by a relative amount 2^{-51} plus an absolute amount of about 2^{-1047}, regardless of whether the operations were exact. The results are therefore valid enclosures, but slightly wider than optimal even for exact operands: the sum of [1, 1] and [1, 1] is an interval of width about 2^{-49} around 2.
di_t di_fast_add(di_t x, di_t y)di_t di_fast_sub(di_t x, di_t y)di_t di_fast_mul(di_t x, di_t y)di_t di_fast_div(di_t x, di_t y)Return an enclosure of the sum, difference, product or quotient of x and y. The product and quotient select the endpoint products by the signs of the operands; a product of the form 0 \cdot \infty gives [-\infty, +\infty], and so does division by an interval containing zero.
di_t di_fast_sqr(di_t x)Returns an enclosure of \{t^2 : t \in x\}. The lower endpoint is nonnegative, and exactly 0 if x contains zero.
di_t di_fast_add_d(di_t x, double y)di_t di_fast_sub_d(di_t x, double y)di_t di_fast_mul_d(di_t x, double y)di_t di_fast_div_d(di_t x, double y)The same operations with the exact point interval [y, y] as the second operand. Throws an exception if y is NaN.
Returns an enclosure of \log x for x \subseteq [0, +\infty]; a nonpositive lower endpoint gives the lower endpoint -\infty. The endpoints are computed with mag_d_log_lower_bound and mag_d_log_upper_bound.
di_t di_fast_mid(di_t x)Returns an enclosure of the midpoint (a + b)/2 of x, or [-\infty, +\infty] if an endpoint is infinite.
double di_fast_ubound_radius(di_t x)Returns an upper bound for the radius (b - a)/2 of x.
Internal functions#
double _di_below(double x)double _di_above(double x)Return a lower respectively upper bound for any real number whose round-to-nearest value is x. _di_below returns x - (|x| + 10^{-300}) \cdot 2^{-51} for x \le 10^{300} (rounded to nearest), 10^{300} for larger x including +\infty, and -\infty for NaN (which arises from \infty - \infty in the endpoint arithmetic); _di_above is symmetric.
The FLINT team, “double_interval.h (module overview),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/double_interval/index.html (generated 2026-10-20).
@misc{flint-doc-double-interval-index,
author = {{The FLINT team}},
title = {{double\_interval.h (module overview)}},
howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
year = {2026},
url = {https://flintlib.org/doc/double_interval/index.html},
note = {Generated 2026-10-20}
}