# double_extras.h – support functions for double arithmetic¶

## Random functions¶

double d_randtest(flint_rand_t state)

Returns a random number in the interval $$[0.5, 1)$$.

double d_randtest_signed(flint_rand_t state, slong minexp, slong maxexp)

Returns a random signed number with exponent between minexp and maxexp or zero.

double d_randtest_special(flint_rand_t state, slong minexp, slong maxexp)

Returns a random signed number with exponent between minexp and maxexp, zero, D_NAN or pmD_INF.

## Arithmetic¶

double d_polyval(const double * poly, int len, double x)

Uses Horner’s rule to evaluate the the polynomial defined by the given len coefficients. Requires that len is nonzero.

## Special functions¶

double d_lambertw(double x)

Computes the principal branch of the Lambert W function, solving the equation $$x = W(x) \exp(W(x))$$. If $$x < -1/e$$, the solution is complex, and NaN is returned.

Depending on the magnitude of $$x$$, we start from a piecewise rational approximation or a zeroth-order truncation of the asymptotic expansion at infinity, and perform 0, 1 or 2 iterations with Halley’s method to obtain full accuracy.

A test of $$10^7$$ random inputs showed a maximum relative error smaller than 0.95 times DBL_EPSILON ($$2^{-52}$$) for positive $$x$$. Accuracy for negative $$x$$ is slightly worse, and can grow to about 10 times DBL_EPSILON close to $$-1/e$$. However, accuracy may be worse depending on compiler flags and the accuracy of the system libm functions.

int d_is_nan(double x)

Returns a nonzero integral value if x is D_NAN, and otherwise returns 0.

double d_log2(double x)

Returns the base 2 logarithm of x provided x is positive. If a domain or pole error occurs, the appropriate error value is returned.