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Recent advances have made numeric debugging tools much faster by using double-double oracles, and numeric analysis tools much more accurate by using condition numbers. But these techniques have downsides: double-double oracles have correlated error so miss floating-point errors while condition numbers cannot cleanly handle over- and underflow. We combine both techniques to avoid these downsides. Our combination, EXPLANIFLOAT, computes condition numbers using double- double arithmetic, which avoids correlated errors. To handle over- and underflow, it introduces a separate logarithmic oracle. As a result, EXPLANIFLOAT achieves a precision of 80.0% and a recall of 96.1% on a collection of 546 difficult numeric benchmarks: more accurate than double-double oracles yet dramatically faster than arbitrary-precision condition number computations. 

An e-graph efficiently represents a congruence relation over many expressions. Although they were originally developed in the late 1970s for use in automated theorem provers, a more recent technique known as equality saturation repurposes e-graphs to implement state-of-the-art, rewrite-driven compiler optimizations and program synthesizers. However, e-graphs remain unspecialized for this newer use case. Equality saturation workloads exhibit distinct characteristics and often require ad-hoc e-graph extensions to incorporate transformations beyond purely syntactic rewrites. This work contributes two techniques that make e-graphs fast and extensible, specializing them to equality saturation. A new amortized invariant restoration technique called rebuilding takes advantage of equality saturation's distinct workload, providing asymptotic speedups over current techniques in practice. A general mechanism called e-class analyses integrates domain-specific analyses into the e-graph, reducing the need for ad hoc manipulation. We implemented these techniques in a new open-source library called egg. Our case studies on three previously published applications of equality saturation highlight how egg's performance and flexibility enable state-of-the-art results across diverse domains. 

Automated verification can ensure that a web page satisfies accessibility, usability, and design properties regardless of the end user's device, preferences, and assistive technologies. However, state-of-the-art verification tools for layout properties do not scale to large pages because they rely on whole-page analyses and must reason about the entire page using the complex semantics of the browser layout algorithm. This paper introduces and formalizes modular layout proofs. A modular layout proof splits a monolithic verification problem into smaller verification problems, one for each component of a web page. Each component specification can use rely/guarantee-style preconditions to make it verifiable independently of the rest of the page and enabling reuse across multiple pages. Modular layout proofs scale verification to pages an order of magnitude larger than those supported by previous approaches. We prototyped these techniques in a new proof assistant, Troika. In Troika, a proof author partitions a page into components and writes specifications for them. Troika then verifies the specifications, and uses those specifications to verify whole-page properties. Troika also enables the proof author to verify different component specifications with different verification tools, leveraging the strengths of each. In a case study, we use Troika to verify a large web page and demonstrate a speed-up of 13--1469x over existing tools, taking verification time from hours to seconds. We develop a systematic approach to writing Troika proofs and demonstrate it on 8 proofs of properties from prior work to show that modular layout proofs are short, easy to write, and provide benefits over existing tools.