[DiadkoShmek]

atan_difference has the correct value at x*y == -1, but its constant selected branch removes the smooth partial derivatives while the inactive singular quotient poisons reverse AD. Both TensorFlow and JAX are affected. This reaches an ordinary finite likelihood:

import tensorflow as tf
import tensorflow_probability as tfp

params = tf.Variable([0., 1.], dtype=tf.float64)  # loc, scale
with tf.GradientTape() as tape:
    lp = tfp.distributions.TruncatedCauchy(
        params[0], params[1], low=-1., high=1.).log_prob(.25)
print(tape.gradient(lp, params))  # currently [NaN, NaN]

The correct score is [8/17, -15/17 + 2/pi], approximately [0.470588, -0.245733]. The density value itself is already correct; forward AD also loses the normalizer's derivative.

Keep the cancellation-resistant quotient away from the pole. At opposite-sign poles, atan(x) - atan(y) is cancellation-free and supplies the correct derivatives. Use the existing native conditional to evaluate that additional work only when a pole is present. The inactive quotient denominator is made finite as well, because JAX vmap may lower conditionals to selects.

Validation:

• Focused regression tests: TensorFlow 2.21 CPU 30 passes, JAX/JAXlib 0.7 CPU 14 passes, versus 20 failures/24 passing controls on BASE. Native regressions cover both dtypes, broadcast adjoints and the public likelihood. The final test also exercises actual TensorFlow ForwardAccumulator through tf.function in four TF2 contexts; its focused BASE comparison fails all four and the candidate passes. Reverse checks remain in all regimes; the TF1 forward limitation is guarded.
• Separate analytic/finite-difference comparisons pass 184 TensorFlow and 184 JAX checks. Independent review passes an unchanged 380-check oracle (108 failures on BASE), plus 72 low/high-scale and mixed/nested-vmap controls. Additional 160 JAX and 60 TensorFlow control comparisons pass; ten NumPy value controls match BASE, without a NumPy AD claim.
• All three affected files pass the unmodified project Pylint configuration with actual Pylint 4.1.3; three existing configuration UserWarnings remain. Seven pre-existing redundant-parenthesis assignments in the already changed owner file are normalized for the whole-file lint gate; those syntax-only changes are AST-equivalent.

Two fixed, warmed, non-pole JAX CPU value-and-gradient fixtures measure scalar: 10.8455 → 11.823 µs (+0.9775µs/+9.0%) and batch of 4,096: 152.184 → 153.5765 µs (+0.9%). These are bounded local cost controls, with compilation recorded separately; they do not establish a general speed or no-regression guarantee.

The numerical scope is finite real inputs and first AD on the tested backends. Full project/Bazel/nightly, GPU, sharding, higher AD and universal IEEE behavior were not tested. The separate Lambert-W change in #2038 is not included.

I directed this work and take responsibility for the contribution and its follow-up. AI assistance was used in producing the change.