def _log_taylor(primals_in, series_in): x, = primals_in series, = series_in u = [x] + series v = [lax.log(x)] + [None] * len(series) for k in range(1, len(v)): conv = sum([_scale(k, j) * v[j] * u[k - j] for j in range(1, k)]) v[k] = (u[k] - fact(k - 1) * conv) / u[0] primal_out, *series_out = v return primal_out, series_out
def _pow_taylor(primals_in, series_in): u_, r_ = primals_in x, series = jet(lambda x, y: lax.mul(y, lax.log(x)), primals_in, series_in) u = [x] + series v = [u_ ** r_] + [None] * len(series) for k in range(1, len(v)): v[k] = fact(k-1) * sum([_scale(k, j)* v[k-j] * u[j] for j in range(1, k+1)]) primal_out, *series_out = v return primal_out, series_out
def_deriv( lax.erf_p, lambda x: lax.mul(lax._const(x, 2. / np.sqrt(np.pi)), lax.exp(lax.neg(lax.square(x))))) def def_comp(prim, comp): """ Define the jet rule for a primitive in terms of a composition of simpler primitives. """ jet_rules[prim] = partial(jet, comp) def_comp(lax.expm1_p, lambda x: lax.exp(x) - 1) def_comp(lax.log1p_p, lambda x: lax.log(1 + x)) def_comp(lax.sqrt_p, lambda x: x**0.5) def_comp(lax.rsqrt_p, lambda x: x**-0.5) def_comp(lax.asinh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) + 1))) def_comp(lax.acosh_p, lambda x: lax.log(x + lax.sqrt(lax.square(x) - 1))) def_comp(lax.atanh_p, lambda x: 0.5 * lax.log(lax.div(1 + x, 1 - x))) def_comp(lax.erfc_p, lambda x: 1 - lax.erf(x)) def_comp(lax.rem_p, lambda x, y: x - y * lax.floor(x / y)) def_comp(lax.clamp_p, lambda a, x, b: lax.min(lax.max(a, x), b)) def _erf_inv_rule(primals_in, series_in): x, = primals_in series, = series_in u = [x] + series