TD Cascade: Triple-Double Precision

Why

Sometimes double-double (~31 decimal digits) is not enough, but quad-double (~64 digits) is more than you need. Triple-double fills the gap: ~47 decimal digits using three doubles, at roughly 75% of the cost of quad-double. This precision tier is particularly useful for intermediate computations where you need more than DD for accumulation accuracy but don’t want to pay the full QD cost.

What

td_cascade wraps floatcascade<3>:

Property	Value
Components	3 doubles
Total bits	192
Significand	~159 bits (~47.8 decimal digits)
Epsilon	~1.39 × 10^-48
Framework	floatcascade<3>

Key Properties

• Triple the precision of double: 159-bit significand
• Intermediate tier: fills the gap between DD (106 bits) and QD (212 bits)
• Unified cascade API: same interface as dd_cascade and qd_cascade
• Memory efficient: 24 bytes vs QD’s 32 bytes
• Same dynamic range as double: 10^-308 to 10^308

Precision Hierarchy

Type	Decimal Digits	Storage	Relative Cost
double	15.9	8 bytes	1x
dd_cascade	31.4	16 bytes	~10x
td_cascade	47.8	24 bytes	~25x
qd_cascade	63.8	32 bytes	~50x

How to Use It

Include

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

Basic Usage

td_cascade a("3.14159265358979323846264338327950288");
td_cascade b("2.71828182845904523536028747135266249");

td_cascade product = a * b;
std::cout << std::setprecision(48) << product << std::endl;
// Accurate to ~47 decimal digits

Extended Accumulation

// When DD accumulation isn't precise enough but QD is overkill
td_cascade sum(0.0);
for (int i = 1; i <= 10000000; ++i) {
    sum += td_cascade(1.0) / td_cascade(i);
}
// Harmonic series sum with ~47 digits of accuracy

Problems It Solves

Problem	How td_cascade Solves It
DD’s 31 digits insufficient, QD’s 64 digits wasteful	47 digits: the right precision for the problem
Need more than DD for accumulation in iterative methods	Triple precision prevents error accumulation
Memory-constrained extended precision	24 bytes vs QD’s 32 bytes (25% savings)
Consistent API across precision levels	Same interface as dd_cascade and qd_cascade