Takum Arithmetic

Takum is a logarithmic tapered-precision number format that uses base √e instead of base 2. It provides superior dynamic range while maintaining precision guarantees that IEEE 754 floats and posits cannot match.

When to Use This Skill

Activate this skill when:

• Implementing Takum numbers in any programming language
• Converting between Takum types and IEEE 754 floats
• Understanding bit-level encoding of Takum values
• Performing arithmetic operations (add, subtract, multiply, divide)
• Implementing transcendental functions (sin, cos, exp, log)
• Comparing Takum with IEEE 754 or posits
• Working with numerical precision requirements
• Designing scientific computing applications

Core Concepts

Two Takum Variants

Variant	Types	Significand Domain	Best For
Logarithmic (standard)	takum_log8/16/32/64	Logarithmic	Multiplication, division, powers
Linear	takum8/16/32/64	Linear	Addition, subtraction

Recommendation: Use logarithmic Takums unless your workload is addition-heavy.

Dynamic Range (Constant for n≥12 bits)

±(√e^-255, √e^255) ≈ ±(4.2×10^-56, 2.4×10^55)

This range is identical for takum16, takum32, and takum64.

Bit Layout

┌─────┬───────────┬─────────────────┬──────────────────────────┐
│  S  │  D │ R₂R₁R₀ │   Characteristic   │       Mantissa        │
│ 1b  │  1b│   3b   │     r bits         │    remaining bits     │
└─────┴───────────┴─────────────────┴──────────────────────────┘

• S (Sign): 0 = positive, 1 = negative
• D (Direction): 0 = magnitude < 1, 1 = magnitude ≥ 1
• R (Regime): Determines characteristic bit count (r = 0 to 7)
• Characteristic: Encodes exponent base √e
• Mantissa: Fractional precision bits

Special Values

Value	Representation
Zero	All bits 0: 0b00000000…
NaR (Not a Real)	Sign=1, rest zeros: 0b10000000…
One	Type-specific positive encoding

Quick Reference

Value Reconstruction

For logarithmic Takum with decoded value l:

value = sign × √e^l = sign × e^(l/2)

Characteristic Ranges

Direction	Characteristic Range
D = 0	c ∈ {-255, …, -1}
D = 1	c ∈ {0, …, 254}

Regime to Mantissa Bits (takum16 example)

Regime r	Mantissa bits p	Characteristic bits
0	11	0
1	10	1
2	9	2
…	…	…
7	4	7

Minimum precision guarantee: At least (n-12) mantissa bits for any value.

Essential Operations

Negation (Two's Complement)

takum_neg(t) = -t  // Standard two's complement negation

Inversion (Logarithmic Takums Only — O(1))

// 1/x is just negating the logarithmic value!
takum_log_inversion(t) = (t ^ 0x7FFF...FF) + 1  // XOR non-sign bits, add 1

Multiplication (Logarithmic — O(1))

// In log domain: multiply = add exponents
l_result = l(a) + l(b)
sign = (a < 0) != (b < 0)

Division (Logarithmic — O(1))

// In log domain: divide = subtract exponents
l_result = l(a) - l(b)
sign = (a < 0) != (b < 0)

Addition/Subtraction

Requires Gaussian logarithm computation — convert to linear domain, compute, convert back.

Implementation Guide

Step 1: Define Types

Use signed integers matching the bit width:

typedef int8_t  takum_log8;
typedef int16_t takum_log16;
typedef int32_t takum_log32;
typedef int64_t takum_log64;

Step 2: Define NaR Constants

NaR is the minimum signed value (two's complement minimum):

#define TAKUM_LOG8_NAR  INT8_MIN   // -128
#define TAKUM_LOG16_NAR INT16_MIN  // -32768
#define TAKUM_LOG32_NAR INT32_MIN  // -2147483648
#define TAKUM_LOG64_NAR INT64_MIN  // -9223372036854775808

Step 3: Create Lookup Tables

// Maps (D|R₂|R₁|R₀) 4-bit index to characteristic bias
static const int16_t c_bias_lut[16] = {
    -255, -127, -63, -31, -15, -7, -3, -1,  // D=0
    0, 1, 3, 7, 15, 31, 63, 127             // D=1
};

// Maps (D|R₂|R₁|R₀) to mantissa bit count for takum16
static const uint8_t p_lut_16[16] = {
    11, 10, 9, 8, 7, 6, 5, 4,  // D=0 (r=0..7)
    11, 10, 9, 8, 7, 6, 5, 4   // D=1 (r=0..7)
};

Step 4: Implement Decoding

// Decode takum_log16 to logarithmic value l
double takum_log16_to_l(takum_log16 t) {
    if (t == TAKUM_LOG16_NAR) return NAN;
    if (t == 0) return -INFINITY;  // log(0) = -∞
    
    bool sign = t < 0;
    uint16_t bits = sign ? -t : t;
    
    uint8_t DR = (bits >> 11) & 0x0F;  // Extract D|R
    int16_t c = c_bias_lut[DR];
    uint8_t p = p_lut_16[DR];
    
    // Extract and add additional characteristic bits
    // Extract mantissa
    // Combine: l = c + m
    
    return sign ? -l : l;
}

Step 5: Implement Encoding

// Encode sign and logarithmic value to takum_log16
takum_log16 takum_log16_from_s_and_l(bool sign, double l) {
    if (isnan(l)) return TAKUM_LOG16_NAR;
    
    // Clamp to representable range
    l = clamp(l, -254.9375, 254.9375);
    
    // Separate into characteristic and mantissa
    int16_t c = (int16_t)floor(fabs(l));
    double m = fabs(l) - c;
    
    // Find DR from c using lookup table
    // Encode bits: S|D|R|C|M
    
    return sign ? -result : result;
}

Conversion Examples

Float64 to Takum

takum_log16 from_float64(double f) {
    if (isnan(f)) return TAKUM_LOG16_NAR;
    if (f == 0.0) return 0;
    
    bool sign = f < 0;
    double l = 2.0 * log(fabs(f));  // l = 2·ln|f| = ln(|f|²)
    
    return takum_log16_from_s_and_l(sign, l);
}

Takum to Float64

double to_float64(takum_log16 t) {
    if (t == TAKUM_LOG16_NAR) return NAN;
    if (t == 0) return 0.0;
    
    double l = takum_log16_to_l(t);
    bool sign = t < 0;
    
    return (sign ? -1.0 : 1.0) * exp(l / 2.0);  // √e^l = e^(l/2)
}

References

For detailed information, see:

• BIT-ENCODING.md — Complete bit layout specification
• MATHEMATICAL-FOUNDATIONS.md — Formal definitions and proofs
• ALGORITHMS.md — Encoding/decoding algorithms
• API-REFERENCE.md — Complete function catalog
• CONVERSION-GUIDE.md — Type conversion patterns
• EDGE-CASES.md — Special value handling
• TESTING-PATTERNS.md — Validation strategies
• BUILD-GUIDE.md — Building the libtakum reference

Key Advantages Over IEEE 754

Property	IEEE 754	Takum
Dynamic range consistency	Varies with precision	Constant for n≥12
Multiplication closure	~25% exact	40%+ exact
Inversion closure	Rare	100% exact (log)
Precision guarantee	Variable	n-12 bits minimum
Zero representation	+0 and -0	Single zero
Special values	NaN, ±Inf, subnormals	Only NaR

Common Patterns

Check for NaR

bool is_nar(takum_log16 t) {
    return t == TAKUM_LOG16_NAR;
}

Safe Division

takum_log16 safe_divide(takum_log16 a, takum_log16 b) {
    if (is_nar(a) || is_nar(b) || b == 0) return TAKUM_LOG16_NAR;
    return takum_log16_division(a, b);
}

Absolute Value (Branchless)

takum_log16 takum_abs(takum_log16 t) {
    return (t < 0) * (-t) + (t >= 0) * t;
}

Transcendental Functions

Use float-domain computation for transcendental functions:

takum_log16 takum_sin(takum_log16 t) {
    double f = to_float64(t);
    double result = sin(f);
    return from_float64(result);
}

For sinpi, cospi variants, use exact values at special angles.

Mathematical Constants

Pre-compute constants for each type:

Constant	takum_log16 value
π	0x3C48
2π	0x4648
√2	0x1684
e	0x2000
ln(2)	0xC91C

See constants.json for all types.