Testing Derivatives and Automatic Differentiation¶

Copyright (C) 2026 Andreas Kloeckner

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In [6]:
import numpy as np
import numpy.linalg as la
In [60]:
def f(xvec):
    x, y = xvec
    return np.array([
        x*y + 2*y**3 - 2,
        x**2*y + 4*y**2*np.cos(x) - 4
        ])
In [64]:
def Jf(xvec):
    x, y = xvec
    return np.array([
        [y, x + 6*y**2],
        [2*x*y - 4*y**2*np.sin(x), x**2 + 8*y*np.cos(x)]
        ])
In [65]:
x = np.random.randn(2)

s = np.random.randn(2)
s /= la.norm(s, 2)
In [67]:
for h in [1e-1, 1e-2, 1e-3, 1e-4, 1e-5]:
    print(h, (f(x + h*s) - f(x))/h - Jf(x)@s)

0.1 [-0.05075743 -0.00600501]
0.01 [-0.00559639 -0.00193633]
0.001 [-0.00056485 -0.00020722]
0.0001 [-5.65366619e-05 -2.08583528e-05]
1e-05 [-5.65420810e-06 -2.08717268e-06]

Now try centered differences.

In [68]:
for h in [1e-1, 1e-2, 1e-3, 1e-4, 1e-5]:
    print(h, (f(x + h*s) - f(x-h*s))/(2*h) - Jf(x)@s)

0.1 [0.00578502 0.01511624]
0.01 [5.78501843e-05 1.51267266e-04]
0.001 [5.78501809e-07 1.51268329e-06]
0.0001 [5.78437953e-09 1.51234321e-08]
1e-05 [4.67468286e-11 1.35421230e-10]

Automatic differentiation (with JAX)¶

In [25]:
import jax.numpy as jnp
from jax import jacfwd, jacrev, make_jaxpr, jvp
In [23]:
def f(xvec):
    x, y = xvec
    return jnp.array([
        x*y + 2*y**3 - 2,
        x**2*y + 4*y**2*jnp.cos(x) - 4
        ])
In [43]:
x = np.random.randn(2)

s = np.random.randn(2)
s /= la.norm(s, 2)

Now subject the JAX-computed Jacobian to the same test as above:

In [42]:
Jf = jacfwd(f)

for h in [1e-1, 1e-2, 1e-3, 1e-4]:
    print(h, (f(x + h*s) - f(x))/h - Jf(x)@s)

0.1 [0.0767622 0.3196597]
0.01 [0.00780439 0.03206491]
0.001 [0.00077128 0.0033834 ]
0.0001 [5.5789948e-05 3.1447411e-03]

Is there a computationally more efficient variant? Consider using

_, jvp_val = jvp(f, (x,), (s,))
In [50]:
for h in [1e-1, 1e-2, 1e-3, 1e-4]:
    _, jvp_val = jvp(f, (x,), (s,))
    print(h, (f(x + h*s) - f(x))/h - jvp_val)

0.1 [-0.00058496  0.26220798]
0.01 [6.3538551e-05 2.3724556e-02]
0.001 [0.00120783 0.00150394]
0.0001 [ 0.01646674 -0.00231075]

How does it work?¶

In [51]:
print(make_jaxpr(f)(jnp.array([1.,2])))

{ lambda ; a:f32[2]. let
    b:f32[1] = slice[limit_indices=(1,) start_indices=(0,) strides=(1,)] a
    c:f32[] = squeeze[dimensions=(0,)] b
    d:f32[1] = slice[limit_indices=(2,) start_indices=(1,) strides=(1,)] a
    e:f32[] = squeeze[dimensions=(0,)] d
    f:f32[] = mul c e
    g:f32[] = integer_pow[y=3] e
    h:f32[] = mul 2.0:f32[] g
    i:f32[] = add f h
    j:f32[] = sub i 2.0:f32[]
    k:f32[] = integer_pow[y=2] c
    l:f32[] = mul k e
    m:f32[] = integer_pow[y=2] e
    n:f32[] = mul 4.0:f32[] m
    o:f32[] = cos c
    p:f32[] = mul n o
    q:f32[] = add l p
    r:f32[] = sub q 4.0:f32[]
    s:f32[1] = broadcast_in_dim[
      broadcast_dimensions=()
      shape=(1,)
      sharding=None
    ] j
    t:f32[1] = broadcast_in_dim[
      broadcast_dimensions=()
      shape=(1,)
      sharding=None
    ] r
    u:f32[2] = concatenate[dimension=0] s t
  in (u,) }
In [52]:
print(make_jaxpr(jacfwd(f))(jnp.array([1.,2])))

{ lambda ; a:f32[2]. let
    b:i32[2,2] = iota[dimension=0 dtype=int32 shape=(2, 2) sharding=None] 
    c:i32[2,2] = iota[dimension=1 dtype=int32 shape=(2, 2) sharding=None] 
    d:i32[2,2] = add b 0:i32[]
    e:bool[2,2] = eq d c
    f:f32[2,2] = convert_element_type[new_dtype=float32 weak_type=False] e
    g:f32[2,2] = split[axis=1 sizes=(np.int64(2),)] f
    h:f32[1] = slice[limit_indices=(1,) start_indices=(0,) strides=(1,)] a
    i:f32[2,1] = slice[limit_indices=(2, 1) start_indices=(0, 0) strides=(1, 1)] g
    j:f32[] = squeeze[dimensions=(0,)] h
    k:f32[2] = squeeze[dimensions=(1,)] i
    l:f32[1] = slice[limit_indices=(2,) start_indices=(1,) strides=(1,)] a
    m:f32[2,1] = slice[limit_indices=(2, 2) start_indices=(0, 1) strides=(1, 1)] g
    n:f32[] = squeeze[dimensions=(0,)] l
    o:f32[2] = squeeze[dimensions=(1,)] m
    p:f32[] = mul j n
    q:f32[2] = mul k n
    r:f32[2] = mul j o
    s:f32[2] = add_any q r
    t:f32[] = integer_pow[y=3] n
    u:f32[] = integer_pow[y=2] n
    v:f32[] = mul 3.0:f32[] u
    w:f32[2] = mul o v
    x:f32[] = mul 2.0:f32[] t
    y:f32[2] = mul 2.0:f32[] w
    z:f32[] = add p x
    ba:f32[2] = add s y
    bb:f32[] = sub z 2.0:f32[]
    bc:f32[] = integer_pow[y=2] j
    bd:f32[] = integer_pow[y=1] j
    be:f32[] = mul 2.0:f32[] bd
    bf:f32[2] = mul k be
    bg:f32[] = mul bc n
    bh:f32[2] = mul bf n
    bi:f32[2] = mul bc o
    bj:f32[2] = add_any bh bi
    bk:f32[] = integer_pow[y=2] n
    bl:f32[] = integer_pow[y=1] n
    bm:f32[] = mul 2.0:f32[] bl
    bn:f32[2] = mul o bm
    bo:f32[] = mul 4.0:f32[] bk
    bp:f32[2] = mul 4.0:f32[] bn
    bq:f32[] = cos j
    br:f32[] = sin j
    bs:f32[2] = mul k br
    bt:f32[2] = neg bs
    bu:f32[] = mul bo bq
    bv:f32[2] = mul bp bq
    bw:f32[2] = mul bo bt
    bx:f32[2] = add_any bv bw
    by:f32[] = add bg bu
    bz:f32[2] = add bj bx
    ca:f32[] = sub by 4.0:f32[]
    cb:f32[1] = broadcast_in_dim[
      broadcast_dimensions=()
      shape=(1,)
      sharding=None
    ] bb
    cc:f32[2,1] = broadcast_in_dim[
      broadcast_dimensions=(0,)
      shape=(2, 1)
      sharding=None
    ] ba
    cd:f32[1] = broadcast_in_dim[
      broadcast_dimensions=()
      shape=(1,)
      sharding=None
    ] ca
    ce:f32[2,1] = broadcast_in_dim[
      broadcast_dimensions=(0,)
      shape=(2, 1)
      sharding=None
    ] bz
    _:f32[2] = concatenate[dimension=0] cb cd
    cf:f32[2,2] = concatenate[dimension=1] cc ce
    cg:f32[2,2] = transpose[permutation=(1, 0)] cf
    ch:f32[2,2] = split[axis=1 sizes=(np.int64(2),)] cg
  in (ch,) }
In [53]:
print(make_jaxpr(jacrev(f))(jnp.array([1.,2])))

{ lambda ; a:f32[2]. let
    b:f32[1] = slice[limit_indices=(1,) start_indices=(0,) strides=(1,)] a
    c:f32[] = squeeze[dimensions=(0,)] b
    d:f32[1] = slice[limit_indices=(2,) start_indices=(1,) strides=(1,)] a
    e:f32[] = squeeze[dimensions=(0,)] d
    f:f32[] = mul c e
    g:f32[] = integer_pow[y=3] e
    h:f32[] = integer_pow[y=2] e
    i:f32[] = mul 3.0:f32[] h
    j:f32[] = mul 2.0:f32[] g
    k:f32[] = add f j
    l:f32[] = sub k 2.0:f32[]
    m:f32[] = integer_pow[y=2] c
    n:f32[] = integer_pow[y=1] c
    o:f32[] = mul 2.0:f32[] n
    p:f32[] = mul m e
    q:f32[] = integer_pow[y=2] e
    r:f32[] = integer_pow[y=1] e
    s:f32[] = mul 2.0:f32[] r
    t:f32[] = mul 4.0:f32[] q
    u:f32[] = cos c
    v:f32[] = sin c
    w:f32[] = mul t u
    x:f32[] = add p w
    y:f32[] = sub x 4.0:f32[]
    z:f32[1] = broadcast_in_dim[
      broadcast_dimensions=()
      shape=(1,)
      sharding=None
    ] l
    ba:f32[1] = broadcast_in_dim[
      broadcast_dimensions=()
      shape=(1,)
      sharding=None
    ] y
    _:f32[2] = concatenate[dimension=0] z ba
    bb:i32[2,2] = iota[dimension=0 dtype=int32 shape=(2, 2) sharding=None] 
    bc:i32[2,2] = iota[dimension=1 dtype=int32 shape=(2, 2) sharding=None] 
    bd:i32[2,2] = add bb 0:i32[]
    be:bool[2,2] = eq bd bc
    bf:f32[2,2] = convert_element_type[new_dtype=float32 weak_type=False] be
    bg:f32[2,2] = split[axis=1 sizes=(np.int64(2),)] bf
    bh:f32[2,1] bi:f32[2,1] = split[axis=1 sizes=(1, 1)] bg
    bj:f32[2] = reduce_sum[axes=(np.int64(1),)] bi
    bk:f32[2] = reduce_sum[axes=(np.int64(1),)] bh
    bl:f32[2] = mul t bj
    bm:f32[2] = mul bj u
    bn:f32[2] = neg bl
    bo:f32[2] = mul bn v
    bp:f32[2] = mul 4.0:f32[] bm
    bq:f32[2] = mul bp s
    br:f32[2] = mul m bj
    bs:f32[2] = add_any bq br
    bt:f32[2] = mul bj e
    bu:f32[2] = mul bt o
    bv:f32[2] = add_any bo bu
    bw:f32[2] = mul 2.0:f32[] bk
    bx:f32[2] = mul bw i
    by:f32[2] = add_any bs bx
    bz:f32[2] = mul c bk
    ca:f32[2] = add_any by bz
    cb:f32[2] = mul bk e
    cc:f32[2] = add_any bv cb
    cd:f32[2,1] = broadcast_in_dim[
      broadcast_dimensions=(0,)
      shape=(2, 1)
      sharding=None
    ] ca
    ce:f32[2,2] = pad[padding_config=((0, 0, 0), (1, np.int64(0), 0))] cd 0.0:f32[]
    cf:f32[2,1] = broadcast_in_dim[
      broadcast_dimensions=(0,)
      shape=(2, 1)
      sharding=None
    ] cc
    cg:f32[2,2] = pad[padding_config=((0, 0, 0), (0, np.int64(1), 0))] cf 0.0:f32[]
    ch:f32[2,2] = add_any ce cg
    ci:f32[2,2] = split[axis=0 sizes=(np.int64(2),)] ch
  in (ci,) }
  • Comment on jacfwd vs jacrev.
  • Comment on jvp vs vjp.
  • Mention jit.
  • Mention vmap.
In [ ]: