DD Cascade: Double-Double via Unified Float Cascade Framework

Why

The classic dd type provides ~106 bits of precision using two doubles. The dd_cascade type achieves the same precision but is built on the unified floatcascade<N> template framework that also underlies td_cascade (triple-double) and qd_cascade (quad-double). This shared architecture means consistent APIs, shared optimizations, and easy switching between precision tiers.

Use dd_cascade when you want double-double precision within the modern cascade framework, or when you need to mix dd, td, and qd precision levels using a consistent interface.

What

dd_cascade wraps floatcascade<2>:

Property	Value
Components	2 doubles
Total bits	128
Significand	~106 bits (~31.4 decimal digits)
Epsilon	~4.93 × 10^-32
Framework	floatcascade<2>

Key Properties

• Unified with td_cascade and qd_cascade: same API, same algorithms, different N
• Identical precision to classic dd: both provide ~106-bit significand
• Array-based access: operator[] for component access
• Fortified arithmetic: volatile modifiers prevent unsafe compiler reordering
• Modern C++20: leverages constexpr and modern template patterns

How to Use It

Include

#include <universal/number/dd_cascade/dd_cascade.hpp>
using namespace sw::universal;

Basic Usage

dd_cascade a(1.0, 1e-17);  // High component + residual
dd_cascade b(2.0, 2e-17);
dd_cascade sum = a + b;

std::cout << to_components(sum) << std::endl;
std::cout << std::setprecision(32) << sum << std::endl;

Precision Tier Switching

// Easy to scale up precision using the same framework
dd_cascade  result_2x(0.0);  // ~31 decimal digits
// td_cascade result_3x(0.0);  // ~47 decimal digits (if needed)
// qd_cascade result_4x(0.0);  // ~64 decimal digits (if needed)

// Same API for all cascade types

When to Use dd_cascade vs dd

Use Case	Recommendation
Standalone double-double computation	Either works; dd is simpler
Mixing dd/td/qd precision tiers	Use cascade family for consistent API
Need component-level array access	Use dd_cascade (operator[])
Integration with new code using cascade framework	Use dd_cascade
Backward compatibility with existing dd code	Use classic dd