use numerics::solve
use numerics::diff
use numerics::fixed_point

# Root finding

fn f1(x) = x³ - x - 2
assert_eq(root_bisect(f1, 1, 2, 1e-10, 1e-10), 1.52137970680, 1e-8)

fn f1_prime(x) = 3 x² - 1
assert_eq(root_newton(f1, f1_prime, 1, 1e-10), 1.52137970680, 1e-8)
assert_eq(root_newton(f1, f1_prime, 2, 1e-10), 1.52137970680, 1e-8)

# Fixed point iteration
let a = 3
fn f_sqrt3(x: Scalar) = 0.5 * (a / x + x)

assert_eq(fixed_point(f_sqrt3, 1, 1e-10), sqrt(3), 1e-10)

# Differentiation

assert_eq(diff(log, 2.0), 0.5, 1e-5)

# TODO: Sadly, the following is not possible at the moment. See https://github.com/sharkdp/numbat/issues/521 for details
# assert_eq(diff(sin, 0.0), 1.0, 1e-5)

assert_eq(diff(sqrt, 1.0), 0.5, 1e-5)

fn f2(x: Scalar) -> Scalar = x² + 4 x + 1

assert_eq(diff(f2, 2.0), 8.0, 1e-5)

fn dist(t: Time) -> Length = 0.5 g0 t^2
fn velocity(t: Time) -> Velocity = diff(dist, t)

assert_eq(velocity(2.0 s), 2.0 s × g0, 1e-3 m/s)