diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index bd0433f..cf7ef05 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -1,12 +1,12 @@
 repos:
   - repo: https://github.com/psf/black
-    rev: stable
+    rev: 22.12.0
     hooks:
     - id: black
-      language_version: python3.9
+      language_version: python3.10
 
   - repo: https://github.com/asottile/reorder_python_imports
-    rev: v2.3.0
+    rev: v3.9.0
     hooks:
     - id: reorder-python-imports
 
diff --git a/jax_cosmo/angular_cl.py b/jax_cosmo/angular_cl.py
index 6db6c9f..1f43c11 100644
--- a/jax_cosmo/angular_cl.py
+++ b/jax_cosmo/angular_cl.py
@@ -11,6 +11,7 @@
 import jax_cosmo.power as power
 import jax_cosmo.transfer as tklib
 from jax_cosmo.scipy.integrate import simps
+from jax_cosmo.scipy.interpolate import InterpolatedUnivariateSpline
 from jax_cosmo.utils import a2z
 from jax_cosmo.utils import z2a
 
@@ -50,7 +51,12 @@ def find_index(a, b):
 
 
 def angular_cl(
-    cosmo, ell, probes, transfer_fn=tklib.Eisenstein_Hu, nonlinear_fn=power.halofit
+    cosmo,
+    ell,
+    probes,
+    transfer_fn=tklib.Eisenstein_Hu,
+    nonlinear_fn=power.halofit,
+    npoints=128,
 ):
     """
     Computes angular Cls for the provided probes
@@ -90,12 +96,17 @@ def combine_kernels(inds):
             # Now kernels has shape [ncls, na]
             kernels = lax.map(combine_kernels, cl_index)
 
-            result = pk * kernels * bkgrd.dchioverda(cosmo, a) / np.clip(chi ** 2, 1.0)
+            result = pk * kernels * bkgrd.dchioverda(cosmo, a) / np.clip(chi**2, 1.0)
 
-            # We transpose the result just to make sure that na is first
-            return result.T
+            return result
 
-        return simps(integrand, z2a(zmax), 1.0, 512) / const.c ** 2
+        atab = np.linspace(z2a(zmax), 1.0, npoints)
+        eval_integral = vmap(
+            lambda x: np.squeeze(
+                InterpolatedUnivariateSpline(atab, x).integral(z2a(zmax), 1.0)
+            )
+        )
+        return eval_integral(integrand(atab)) / const.c**2
 
     return cl(ell)
 
diff --git a/jax_cosmo/background.py b/jax_cosmo/background.py
index 1a2a182..ba7566c 100644
--- a/jax_cosmo/background.py
+++ b/jax_cosmo/background.py
@@ -291,7 +291,7 @@ def dchioverda(cosmo, a):
 
         \frac{d \chi}{da}(a) = \frac{R_H}{a^2 E(a)}
     """
-    return const.rh / (a ** 2 * np.sqrt(Esqr(cosmo, a)))
+    return const.rh / (a**2 * np.sqrt(Esqr(cosmo, a)))
 
 
 def transverse_comoving_distance(cosmo, a):
diff --git a/jax_cosmo/power.py b/jax_cosmo/power.py
index 4d28cb1..62dcfb7 100644
--- a/jax_cosmo/power.py
+++ b/jax_cosmo/power.py
@@ -16,7 +16,7 @@ def primordial_matter_power(cosmo, k):
     """Primordial power spectrum
     Pk = k^n
     """
-    return k ** cosmo.n_s
+    return k**cosmo.n_s
 
 
 def linear_matter_power(cosmo, k, a=1.0, transfer_fn=tklib.Eisenstein_Hu, **kwargs):
@@ -45,9 +45,9 @@ def linear_matter_power(cosmo, k, a=1.0, transfer_fn=tklib.Eisenstein_Hu, **kwar
     g = bkgrd.growth_factor(cosmo, a)
     t = transfer_fn(cosmo, k, **kwargs)
 
-    pknorm = cosmo.sigma8 ** 2 / sigmasqr(cosmo, 8.0, transfer_fn, **kwargs)
+    pknorm = cosmo.sigma8**2 / sigmasqr(cosmo, 8.0, transfer_fn, **kwargs)
 
-    pk = primordial_matter_power(cosmo, k) * t ** 2 * g ** 2
+    pk = primordial_matter_power(cosmo, k) * t**2 * g**2
 
     # Apply normalisation
     pk = pk * pknorm
@@ -76,7 +76,7 @@ def int_sigma(logk):
         return k * (k * w) ** 2 * pk
 
     y = romb(int_sigma, np.log10(kmin), np.log10(kmax), divmax=7)
-    return 1.0 / (2.0 * np.pi ** 2.0) * y
+    return 1.0 / (2.0 * np.pi**2.0) * y
 
 
 def linear(cosmo, k, a, transfer_fn):
@@ -103,10 +103,10 @@ def int_sigma(logk):
             pk = linear_matter_power(cosmo, k, transfer_fn=transfer_fn)
             g = bkgrd.growth_factor(cosmo, np.atleast_1d(a))
             return (
-                np.expand_dims(pk * k ** 3, axis=1)
-                * np.exp(-(y ** 2))
-                / (2.0 * np.pi ** 2)
-                * g ** 2
+                np.expand_dims(pk * k**3, axis=1)
+                * np.exp(-(y**2))
+                / (2.0 * np.pi**2)
+                * g**2
             )
 
         sigma = simps(int_sigma, np.log(1e-4), np.log(1e4), 256)
@@ -125,13 +125,13 @@ def integrand(logk):
         pk = linear_matter_power(cosmo, k, transfer_fn=transfer_fn)
         g = np.expand_dims(bkgrd.growth_factor(cosmo, np.atleast_1d(a)), 0)
         res = (
-            np.expand_dims(pk * k ** 3, axis=1)
-            * np.exp(-(y ** 2))
-            * g ** 2
-            / (2.0 * np.pi ** 2)
+            np.expand_dims(pk * k**3, axis=1)
+            * np.exp(-(y**2))
+            * g**2
+            / (2.0 * np.pi**2)
         )
-        dneff_dlogk = 2 * res * y ** 2
-        dC_dlogk = 4 * res * (y ** 2 - y ** 4)
+        dneff_dlogk = 2 * res * y**2
+        dC_dlogk = 4 * res * (y**2 - y**4)
         return np.stack([dneff_dlogk, dC_dlogk], axis=1)
 
     res = simps(integrand, np.log(1e-4), np.log(1e4), 256)
@@ -185,44 +185,44 @@ def halofit(cosmo, k, a, transfer_fn, prescription="takahashi2012"):
         a_n = 10 ** (
             1.4861
             + 1.8369 * n
-            + 1.6762 * n ** 2
-            + 0.7940 * n ** 3
-            + 0.1670 * n ** 4
+            + 1.6762 * n**2
+            + 0.7940 * n**3
+            + 0.1670 * n**4
             - 0.6206 * C
         )
-        b_n = 10 ** (0.9463 + 0.9466 * n + 0.3084 * n ** 2 - 0.9400 * C)
-        c_n = 10 ** (-0.2807 + 0.6669 * n + 0.3214 * n ** 2 - 0.0793 * C)
+        b_n = 10 ** (0.9463 + 0.9466 * n + 0.3084 * n**2 - 0.9400 * C)
+        c_n = 10 ** (-0.2807 + 0.6669 * n + 0.3214 * n**2 - 0.0793 * C)
         gamma_n = 0.8649 + 0.2989 * n + 0.1631 * C
-        alpha_n = 1.3884 + 0.3700 * n - 0.1452 * n ** 2
-        beta_n = 0.8291 + 0.9854 * n + 0.3401 * n ** 2
+        alpha_n = 1.3884 + 0.3700 * n - 0.1452 * n**2
+        beta_n = 0.8291 + 0.9854 * n + 0.3401 * n**2
         mu_n = 10 ** (-3.5442 + 0.1908 * n)
         nu_n = 10 ** (0.9585 + 1.2857 * n)
     elif prescription == "takahashi2012":
         a_n = 10 ** (
             1.5222
             + 2.8553 * n
-            + 2.3706 * n ** 2
-            + 0.9903 * n ** 3
-            + 0.2250 * n ** 4
+            + 2.3706 * n**2
+            + 0.9903 * n**3
+            + 0.2250 * n**4
             - 0.6038 * C
             + 0.1749 * om_de * (1 + w)
         )
         b_n = 10 ** (
             -0.5642
             + 0.5864 * n
-            + 0.5716 * n ** 2
+            + 0.5716 * n**2
             - 1.5474 * C
             + 0.2279 * om_de * (1 + w)
         )
-        c_n = 10 ** (0.3698 + 2.0404 * n + 0.8161 * n ** 2 + 0.5869 * C)
+        c_n = 10 ** (0.3698 + 2.0404 * n + 0.8161 * n**2 + 0.5869 * C)
         gamma_n = 0.1971 - 0.0843 * n + 0.8460 * C
-        alpha_n = np.abs(6.0835 + 1.3373 * n - 0.1959 * n ** 2 - 5.5274 * C)
+        alpha_n = np.abs(6.0835 + 1.3373 * n - 0.1959 * n**2 - 5.5274 * C)
         beta_n = (
             2.0379
             - 0.7354 * n
-            + 0.3157 * n ** 2
-            + 1.2490 * n ** 3
-            + 0.3980 * n ** 4
+            + 0.3157 * n**2
+            + 1.2490 * n**3
+            + 0.3980 * n**4
             - 0.1682 * C
         )
         mu_n = 0.0
@@ -232,7 +232,7 @@ def halofit(cosmo, k, a, transfer_fn, prescription="takahashi2012"):
 
     f1a = om_m ** (-0.0732)
     f2a = om_m ** (-0.1423)
-    f3a = om_m ** 0.0725
+    f3a = om_m**0.0725
     f1b = om_m ** (-0.0307)
     f2b = om_m ** (-0.0585)
     f3b = om_m ** (0.0743)
@@ -248,21 +248,21 @@ def halofit(cosmo, k, a, transfer_fn, prescription="takahashi2012"):
     else:
         raise NotImplementedError
 
-    f = lambda x: x / 4.0 + x ** 2 / 8.0
+    f = lambda x: x / 4.0 + x**2 / 8.0
 
-    d2l = k ** 3 * pklin / (2.0 * np.pi ** 2)
+    d2l = k**3 * pklin / (2.0 * np.pi**2)
 
     y = k / k_nl
 
     # Eq C2
     d2q = d2l * ((1.0 + d2l) ** beta_n / (1 + alpha_n * d2l)) * np.exp(-f(y))
     d2hprime = (
-        a_n * y ** (3 * f1) / (1.0 + b_n * y ** f2 + (c_n * f3 * y) ** (3.0 - gamma_n))
+        a_n * y ** (3 * f1) / (1.0 + b_n * y**f2 + (c_n * f3 * y) ** (3.0 - gamma_n))
     )
-    d2h = d2hprime / (1.0 + mu_n / y + nu_n / y ** 2)
+    d2h = d2hprime / (1.0 + mu_n / y + nu_n / y**2)
     # Eq. C1
     d2nl = d2q + d2h
-    pk_nl = 2.0 * np.pi ** 2 / k ** 3 * d2nl
+    pk_nl = 2.0 * np.pi**2 / k**3 * d2nl
 
     return pk_nl.squeeze()
 
diff --git a/jax_cosmo/probes.py b/jax_cosmo/probes.py
index fe47895..fab6880 100644
--- a/jax_cosmo/probes.py
+++ b/jax_cosmo/probes.py
@@ -70,7 +70,7 @@ def integrand_single(z_prime):
     radial_kernel = radial_kernel[inv]
 
     # Constant term
-    constant_factor = 3.0 * const.H0 ** 2 * cosmo.Omega_m / 2.0 / const.c
+    constant_factor = 3.0 * const.H0**2 * cosmo.Omega_m / 2.0 / const.c
     # Ell dependent factor
     ell_factor = np.sqrt((ell - 1) * (ell) * (ell + 1) * (ell + 2)) / (ell + 0.5) ** 2
     return constant_factor * ell_factor * radial_kernel
@@ -222,7 +222,7 @@ def noise(self):
             sigma_e = np.array([s for s in self.config["sigma_e"]])
         else:
             sigma_e = self.config["sigma_e"]
-        return sigma_e ** 2 / ngals
+        return sigma_e**2 / ngals
 
 
 @register_pytree_node_class
diff --git a/jax_cosmo/redshift.py b/jax_cosmo/redshift.py
index e02e9c7..c7fb4b2 100644
--- a/jax_cosmo/redshift.py
+++ b/jax_cosmo/redshift.py
@@ -75,7 +75,7 @@ class smail_nz(redshift_distribution):
 
     def pz_fn(self, z):
         a, b, z0 = self.params
-        return z ** a * np.exp(-((z / z0) ** b))
+        return z**a * np.exp(-((z / z0) ** b))
 
 
 @register_pytree_node_class
@@ -118,7 +118,7 @@ class kde_nz(redshift_distribution):
     def _kernel(self, bw, X, x):
         """Gaussian kernel for KDE"""
         return (1.0 / np.sqrt(2 * np.pi) / bw) * np.exp(
-            -((X - x) ** 2) / (bw ** 2 * 2.0)
+            -((X - x) ** 2) / (bw**2 * 2.0)
         )
 
     def pz_fn(self, z):
diff --git a/jax_cosmo/scipy/integrate.py b/jax_cosmo/scipy/integrate.py
index f19dae3..baca247 100644
--- a/jax_cosmo/scipy/integrate.py
+++ b/jax_cosmo/scipy/integrate.py
@@ -61,7 +61,7 @@ def _romberg_diff(b, c, k):
     Compute the differences for the Romberg quadrature corrections.
     See Forman Acton's "Real Computing Made Real," p 143.
     """
-    tmp = 4.0 ** k
+    tmp = 4.0**k
     return (tmp * c - b) / (tmp - 1.0)
 
 
@@ -143,7 +143,7 @@ def scan_fn(carry, y):
         return (x, k + 1), x
 
     for i in range(1, divmax + 1):
-        n = 2 ** i
+        n = 2**i
         ordsum = ordsum + _difftrapn(vfunc, interval, n)
 
         x = intrange * ordsum / n
diff --git a/jax_cosmo/scipy/interpolate.py b/jax_cosmo/scipy/interpolate.py
index 9c34e7a..d1389de 100644
--- a/jax_cosmo/scipy/interpolate.py
+++ b/jax_cosmo/scipy/interpolate.py
@@ -258,11 +258,11 @@ def __call__(self, x):
 
         if self.k == 2:
             t, a, b, c = self._compute_coeffs(x)
-            result = a + b * t + c * t ** 2
+            result = a + b * t + c * t**2
 
         if self.k == 3:
             t, a, b, c, d = self._compute_coeffs(x)
-            result = a + b * t + c * t ** 2 + d * t ** 3
+            result = a + b * t + c * t**2 + d * t**3
 
         return result
 
@@ -300,7 +300,7 @@ def _compute_coeffs(self, xs):
             dt = (x - knots[:-1])[ind]
             b = coefficients[ind]
             b1 = coefficients[ind + 1]
-            a = y[ind] - b * dt - (b1 - b) * dt ** 2 / (2 * h)
+            a = y[ind] - b * dt - (b1 - b) * dt**2 / (2 * h)
             c = (b1 - b) / (2 * h)
             result = (t, a, b, c)
 
@@ -343,7 +343,7 @@ def derivative(self, x, n=1):
             if self.k == 3:
                 t, a, b, c, d = self._compute_coeffs(x)
                 if n == 1:
-                    result = b + 2 * c * t + 3 * d * t ** 2
+                    result = b + 2 * c * t + 3 * d * t**2
                 if n == 2:
                     result = 2 * c + 6 * d * t
                 if n == 3:
@@ -382,20 +382,20 @@ def antiderivative(self, xs):
             a = y[:-1]
             b = coefficients
             h = np.diff(knots)
-            cst = np.concatenate([np.zeros(1), np.cumsum(a * h + b * h ** 2 / 2)])
-            return cst[ind] + a[ind] * t + b[ind] * t ** 2 / 2
+            cst = np.concatenate([np.zeros(1), np.cumsum(a * h + b * h**2 / 2)])
+            return cst[ind] + a[ind] * t + b[ind] * t**2 / 2
 
         if self.k == 2:
             h = np.diff(knots)
             dt = x - knots[:-1]
             b = coefficients[:-1]
             b1 = coefficients[1:]
-            a = y - b * dt - (b1 - b) * dt ** 2 / (2 * h)
+            a = y - b * dt - (b1 - b) * dt**2 / (2 * h)
             c = (b1 - b) / (2 * h)
             cst = np.concatenate(
-                [np.zeros(1), np.cumsum(a * h + b * h ** 2 / 2 + c * h ** 3 / 3)]
+                [np.zeros(1), np.cumsum(a * h + b * h**2 / 2 + c * h**3 / 3)]
             )
-            return cst[ind] + a[ind] * t + b[ind] * t ** 2 / 2 + c[ind] * t ** 3 / 3
+            return cst[ind] + a[ind] * t + b[ind] * t**2 / 2 + c[ind] * t**3 / 3
 
         if self.k == 3:
             h = np.diff(knots)
@@ -408,15 +408,15 @@ def antiderivative(self, xs):
             cst = np.concatenate(
                 [
                     np.zeros(1),
-                    np.cumsum(a * h + b * h ** 2 / 2 + c * h ** 3 / 3 + d * h ** 4 / 4),
+                    np.cumsum(a * h + b * h**2 / 2 + c * h**3 / 3 + d * h**4 / 4),
                 ]
             )
             return (
                 cst[ind]
                 + a[ind] * t
-                + b[ind] * t ** 2 / 2
-                + c[ind] * t ** 3 / 3
-                + d[ind] * t ** 4 / 4
+                + b[ind] * t**2 / 2
+                + c[ind] * t**3 / 3
+                + d[ind] * t**4 / 4
             )
 
     def integral(self, a, b):
@@ -440,3 +440,32 @@ def integral(self, a, b):
             sign = -1
         xs = np.array([a, b])
         return sign * np.diff(self.antiderivative(xs))
+
+
+def splint(func, a, b, k=3, N=128):
+    """Function that computes an integration with a spline function
+    slightly different from the original splint from scipy
+    """
+    x = np.linspace(a, b, N)
+    return InterpolatedUnivariateSpline(x, func(x), k=k).integral(a, b)
+
+
+# def splint_fwd(func, a, b, **kwargs):
+#     result = splint(func, a, b, **kwargs)
+#     aux = (a, b, kwargs)
+#     return result, aux
+
+# def splint_bwd(func, aux, grad):
+#     a, b, kwargs = aux
+
+#     grad_a = -grad * func(a)
+#     grad_b = grad * func(b)
+
+#     grad_args = []
+#     for i in range(len(args)):
+#         def _vjp_func(_t, *_args):
+#             return jax.grad(func, i)(_t, *_args)
+#         grad_args.append(grad * quad(_vjp_func, a, b, args))
+#     grad_args = tuple(grad_args)
+
+#     return grad_a, grad_b, grad_args
diff --git a/jax_cosmo/transfer.py b/jax_cosmo/transfer.py
index 23def59..ce66c80 100644
--- a/jax_cosmo/transfer.py
+++ b/jax_cosmo/transfer.py
@@ -45,7 +45,7 @@ def Eisenstein_Hu(cosmo, k, type="eisenhu_osc"):
     #            - sh_d   : sound horizon at drag epoch
     #            - k_silk : Silk damping scale
     T_2_7_sqr = (const.tcmb / 2.7) ** 2
-    h2 = cosmo.h ** 2
+    h2 = cosmo.h**2
     w_m = cosmo.Omega_m * h2
     w_b = cosmo.Omega_b * h2
     fb = cosmo.Omega_b / cosmo.Omega_m
@@ -111,7 +111,7 @@ def Eisenstein_Hu(cosmo, k, type="eisenhu_osc"):
         # EH98 (11, 12)
         a1 = np.power(46.9 * w_m, 0.670) * (1.0 + np.power(32.1 * w_m, -0.532))
         a2 = np.power(12.0 * w_m, 0.424) * (1.0 + np.power(45.0 * w_m, -0.582))
-        alpha_c = np.power(a1, -fb) * np.power(a2, -(fb ** 3))
+        alpha_c = np.power(a1, -fb) * np.power(a2, -(fb**3))
         b1 = 0.944 / (1.0 + np.power(458.0 * w_m, -0.708))
         b2 = np.power(0.395 * w_m, -0.0266)
         beta_c = 1.0 + b1 * (np.power(fc, b2) - 1.0)
diff --git a/tests/test_power.py b/tests/test_power.py
index 3c49939..8e08ed4 100644
--- a/tests/test_power.py
+++ b/tests/test_power.py
@@ -38,7 +38,7 @@ def test_eisenstein_hu():
     # Computing matter power spectrum
     pk_ccl = ccl.linear_matter_power(cosmo_ccl, k, 1.0)
     pk_jax = (
-        power.linear_matter_power(cosmo_jax, k / cosmo_jax.h, a=1.0) / cosmo_jax.h ** 3
+        power.linear_matter_power(cosmo_jax, k / cosmo_jax.h, a=1.0) / cosmo_jax.h**3
     )
 
     assert_allclose(pk_ccl, pk_jax, rtol=0.5e-2)
@@ -81,7 +81,7 @@ def test_halofit():
             transfer_fn=tklib.Eisenstein_Hu,
             nonlinear_fn=power.halofit,
         )
-        / cosmo_jax.h ** 3
+        / cosmo_jax.h**3
     )
 
     assert_allclose(pk_ccl, pk_jax, rtol=0.5e-2)
@@ -130,7 +130,7 @@ def test_halofit_nl_scales():
             transfer_fn=tklib.Eisenstein_Hu,
             nonlinear_fn=power.halofit,
         )
-        / cosmo_jax.h ** 3
+        / cosmo_jax.h**3
     )
     assert_allclose(pk_ccl, pk_jax, rtol=0.5e-2)
 
@@ -145,7 +145,7 @@ def test_halofit_nl_scales():
             transfer_fn=tklib.Eisenstein_Hu,
             nonlinear_fn=power.halofit,
         )
-        / cosmo_jax.h ** 3
+        / cosmo_jax.h**3
     )
     # We relax the test here, because actually CCL is not accurate in this regime
     assert_allclose(pk_ccl, pk_jax, rtol=2e-2)