FR_Math 2.0.5

FR_Math Release Notes

Version 2.0.5 (2026)

Release pipeline fixes. No functional changes to the math library.

• Fixed tools/make_release.sh failing after squash-merge: local master
  diverges from origin (squash creates a new commit), so git pull --ff-only fails. Now detects divergence and resets to origin/master.
• Fixed on-master release path: script now pushes master to origin and
  waits for CI before tagging (previously skipped both, causing tags to
  point at commits not yet on the remote).
• Release pipeline auto-commits pipeline-generated changes (badge
  updates, version sync) instead of failing on a dirty working tree.
  Unexpected dirty files still block the release.

Version 2.0.4 (2026)

CI fix release. No functional changes to the math library.

• Fixed release.yml coverage step failing due to stale gcov invocation
• Removed conflicting auto-release job from ci.yml (replaced by
  tag-triggered release.yml)
• Documentation updated: release_management.md, docs/building.md,
  and pages/guide/building.html now accurately describe the guided
  release pipeline

---

Version 2.0.3 (2026)

CI and release pipeline cleanup. No functional changes to the math
library itself.

Release pipeline

• New guided release script tools/make_release.sh (ported from xelp):
  validates, opens PR, waits for CI, merges, tags, waits for GitHub
  Release, then publishes to PlatformIO and ESP-IDF registries
• Removed old scripts/make_release.sh (validation-only, manual merge)
• All doc/page references updated to tools/make_release.sh
• Cleaned up stale branches

---

Version 2.0.2 (2026)

Embedded library publishing support. No functional changes to the
math library itself — this release adds package manager integration
and improves example organization.

Library publishing

• Added library.properties for Arduino Library Manager
• Added library.json for PlatformIO Registry
• Added idf_component.yml + CMakeLists.txt for ESP-IDF Component Registry
• Added keywords.txt for Arduino IDE syntax highlighting
• Added .github/workflows/release.yml for tag-triggered release validation

Examples reorganized

• New focused Arduino sketches: basic-math, trig-functions, wave-generators
• Moved FR_Math_Example1.cpp to examples/posix-example/
• Removed symlinks from examples/arduino_smoke/ (Arduino forbids symlinks)
• Added examples/README.md

Documentation

• Added llms.txt — machine-readable API summary for AI coding agents
• Added agents.md — coding agent conventions and contribution guide
• Added dev/misc/ publish checklists for Arduino, PlatformIO, and ESP-IDF
• Updated scripts/sync_version.sh to sync version across all new metadata files
• Added release step 11: verify llms.txt and agents.md are current

---

Version 2.0.1 (2026)

Precision and accuracy release. All changes are backward-compatible
with v2.0.0 except where noted.

Trig output precision

• fr_cos_bam / fr_sin_bam now return s15.16 (was s0.15).
  Exact values at cardinal angles: cos(0°) = 65536, cos(90°) = 0.
  FR_TRIG_ONE = 65536.
• FR_TanI / fr_tan return s15.16 (was s16.15). Saturation
  is now ±INT32_MAX.
• All wrappers updated: FR_Cos, FR_Sin, FR_CosI, FR_SinI,
  FR_TanI, FR_Tan.

Inverse trig now returns radians

• FR_acos, FR_asin, FR_atan gain an out_radix parameter
  and return radians at that radix (was degrees as s16).
• FR_atan2(y, x, out_radix) also returns radians.
• FR_BAM2RAD macro corrected (was off by a factor of 1024).

Rounding improvements

• FR_FixMuls / FR_FixMulSat: add 0.5 LSB (+0x8000) before
  the >>16 shift. Both now round to nearest instead of
  truncating.
• FR_sqrt / FR_hypot: the internal fr_isqrt64 now rounds to
  nearest (remainder > root → +1). Worst-case error drops from
  <1 LSB to ≤ 0.5 LSB.
• FR_DIV now rounds to nearest (≤ 0.5 LSB error) instead of
  truncating. FR_DIV_TRUNC preserves the old truncating behaviour.

Improved exp / log accuracy

• FR_pow2 table expanded from 17 entries (16 segments) to
  65 entries (64 segments, 260 bytes). Interpolation error drops
  by ~16×.
• FR_log2 table expanded from 33 entries to 65 entries
  (6-bit index / 24-bit interpolation). Worst-case error ≤ 4 LSB
  at Q16.16.
• FR_MULK28 macro added: multiplies any fixed-point value
  by a radix-28 constant using a 64-bit intermediate with
  round-to-nearest. ~9 decimal digits of precision.
• FR_EXP and FR_POW10 now use FR_MULK28 for the base
  conversion instead of shift-only macros.
• FR_ln and FR_log10 also use FR_MULK28 internally.
• FR_EXP_FAST and FR_POW10_FAST added as shift-only
  alternatives for 8-bit targets where 64-bit multiply is
  expensive.

New symbols

• FR_MIN, FR_MAX, FR_CLAMP — standard min/max/clamp.
• FR_DIV_TRUNC(x, xr, y, yr) — truncating division (the old
  FR_DIV behaviour).
• FR_MOD(x, xr, y, yr) — fixed-point modulus.
• Radix-28 constants: FR_kLOG2E_28, FR_krLOG2E_28,
  FR_kLOG2_10_28, FR_krLOG2_10_28.

Breaking changes from v2.0.0

Change	v2.0.0	v2.0.1
sin/cos return type	s16 (s0.15)	s32 (s15.16)
sin/cos 1.0 value	32767	65536 (exact)
tan return format	s16.15 (radix 15)	s15.16 (radix 16)
tan saturation	±(32767 << 15)	±INT32_MAX
FR_acos/asin signature	(input, radix) → s16 degrees	(input, radix, out_radix) → s32 radians
FR_atan signature	(input, radix) → s16 degrees	(input, radix, out_radix) → s32 radians
FR_atan2 signature	(y, x) → s16 degrees	(y, x, out_radix) → s32 radians
FR_BAM2RAD	off by 1024× (bug)	correct
FR_DIV rounding	truncates toward zero	rounds to nearest (use FR_DIV_TRUNC for old behaviour)

---

Version 2.0.0 (2026)

This is a significant bug-fix and precision-improvement release. Several
v1 functions produced wrong numerical results on every platform due to
arithmetic bugs; others produced wrong results specifically on 64-bit
hosts because of a typedef choice. Every changed function is covered by
the TDD characterization suite in tests/test_tdd.cpp.

See dev/fr_math_precision.md for the full per-symbol reference
(inputs, outputs, precision, saturation) and dev/fr_math2_impl_plan.md
for the implementation plan this release executed.

64-bit safety (portability)

• FR_defs.h: migrated typedefs to <stdint.h> (s8 → int8_t,
  s32 → int32_t, etc.). v1 defined s32 as signed long, which is
  64 bits on LP64 platforms (Linux and macOS on x64 / ARM64), so every
  fixed-point computation that relied on s32 being exactly 32 bits
  silently produced wrong answers on desktop Linux and macOS. C99
  <stdint.h> is now mandatory in v2 (every modern toolchain — gcc,
  clang, MSVC, IAR, Keil C51, sdcc, MSP430-gcc, AVR-gcc, RISC-V/ARM —
  ships it). If you are stuck on a pre-C99 compiler, FR_Math 1.0.x
  remains the version for you.

Numerical fixes

• FR_FixMulSat / FR_FixMuls: v1 used a split-multiply formula
  with an algebraic bug that returned wrong values for certain sign
  combinations. v2 uses an int64_t fast path with explicit
  saturation.
• FR_log2: v1 was missing the accumulator in the mantissa-table
  step and returned wrong values for non-power-of-2 inputs. v2 rewrites
  the algorithm: leading-bit-position → normalize to s1.30 → 33-entry
  mantissa lookup with linear interpolation.
• FR_ln / FR_log10: inherit the FR_log2 fix automatically
  (they are multiplies of FR_log2 by a constant).
• FR_pow2: v1 used C truncation toward zero to extract the integer
  exponent, which gave wrong answers for negative non-integer inputs
  (e.g. FR_pow2(-0.5) returned ~2 instead of ~0.707). v2 uses
  mathematical floor (toward −∞) and a 17-entry fraction table with
  linear interp.
• FR_atan2: v1 was a placeholder that returned garbage. v2 is a
  correct octant-reduced arctan with a 33-entry table. Max error ≤ 1°.
  v2 also drops the vestigial radix parameter — new signature is
  FR_atan2(s32 y, s32 x).
• FR_atan: v1 was wired up incorrectly. v2 implements as
  FR_atan2(input, 1<<radix).
• FR_Tan: v1 used s16 loop variables that overflowed for angles
  near 90°. v2 uses s32.
• FR_TanI: removed dead unreachable code (if (270 == deg)).
• FR_printNumD: v1 invoked unary minus on INT_MIN, which is
  undefined behavior, and always returned 0. v2 works in unsigned
  magnitude and returns the real byte count (or −1 on null f).
• FR_printNumF: same INT_MIN fix, plus v1's fraction extraction
  was mathematically wrong and printed bogus tail digits.
• FR_printNumH: v1 shifted a signed negative int right. v2 casts
  to unsigned first.

Macro fixes

• FR_DEG2RAD and FR_RAD2DEG: v1 had the macro bodies swapped
  relative to their names, so FR_DEG2RAD(x) actually multiplied by
  57.3 and FR_RAD2DEG(x) multiplied by 0.017. v2 swaps them back.
  FR_DEG2RAD also had missing parens around x in a subexpression.
• FR_NUM: v1 signature was FR_NUM(i, f, r) and the body ignored
  the f argument entirely. v2 adds an explicit d digit-count
  argument — new signature is FR_NUM(i, f, d, r). Any caller of the
  v1 form must be updated.

New functionality

• Radian-native trig (fr_cos, fr_sin, fr_tan, fr_cos_bam,
  fr_sin_bam, fr_cos_deg, fr_sin_deg): takes radians (or BAM, or
  integer degrees) directly rather than forcing everything through
  integer degrees. Uses a new 129-entry s0.15 quadrant cosine table in
  src/FR_trig_table.h with linear interpolation and...

FR_Math 2.0.2

FR_Math Release Notes

Version 2.0.2 (2026)

Embedded library publishing support. No functional changes to the
math library itself — this release adds package manager integration
and improves example organization.

Library publishing

• Added library.properties for Arduino Library Manager
• Added library.json for PlatformIO Registry
• Added idf_component.yml + CMakeLists.txt for ESP-IDF Component Registry
• Added keywords.txt for Arduino IDE syntax highlighting
• Added .github/workflows/release.yml for tag-triggered release validation

Examples reorganized

• New focused Arduino sketches: basic-math, trig-functions, wave-generators
• Moved FR_Math_Example1.cpp to examples/posix-example/
• Removed symlinks from examples/arduino_smoke/ (Arduino forbids symlinks)
• Added examples/README.md

Documentation

• Added llms.txt — machine-readable API summary for AI coding agents
• Added agents.md — coding agent conventions and contribution guide
• Added dev/misc/ publish checklists for Arduino, PlatformIO, and ESP-IDF
• Updated scripts/sync_version.sh to sync version across all new metadata files
• Added release step 11: verify llms.txt and agents.md are current

---

Version 2.0.1 (2026)

Precision and accuracy release. All changes are backward-compatible
with v2.0.0 except where noted.

Trig output precision

• fr_cos_bam / fr_sin_bam now return s15.16 (was s0.15).
  Exact values at cardinal angles: cos(0°) = 65536, cos(90°) = 0.
  FR_TRIG_ONE = 65536.
• FR_TanI / fr_tan return s15.16 (was s16.15). Saturation
  is now ±INT32_MAX.
• All wrappers updated: FR_Cos, FR_Sin, FR_CosI, FR_SinI,
  FR_TanI, FR_Tan.

Inverse trig now returns radians

• FR_acos, FR_asin, FR_atan gain an out_radix parameter
  and return radians at that radix (was degrees as s16).
• FR_atan2(y, x, out_radix) also returns radians.
• FR_BAM2RAD macro corrected (was off by a factor of 1024).

Rounding improvements

• FR_FixMuls / FR_FixMulSat: add 0.5 LSB (+0x8000) before
  the >>16 shift. Both now round to nearest instead of
  truncating.
• FR_sqrt / FR_hypot: the internal fr_isqrt64 now rounds to
  nearest (remainder > root → +1). Worst-case error drops from
  <1 LSB to ≤ 0.5 LSB.
• FR_DIV now rounds to nearest (≤ 0.5 LSB error) instead of
  truncating. FR_DIV_TRUNC preserves the old truncating behaviour.

Improved exp / log accuracy

• FR_pow2 table expanded from 17 entries (16 segments) to
  65 entries (64 segments, 260 bytes). Interpolation error drops
  by ~16×.
• FR_log2 table expanded from 33 entries to 65 entries
  (6-bit index / 24-bit interpolation). Worst-case error ≤ 4 LSB
  at Q16.16.
• FR_MULK28 macro added: multiplies any fixed-point value
  by a radix-28 constant using a 64-bit intermediate with
  round-to-nearest. ~9 decimal digits of precision.
• FR_EXP and FR_POW10 now use FR_MULK28 for the base
  conversion instead of shift-only macros.
• FR_ln and FR_log10 also use FR_MULK28 internally.
• FR_EXP_FAST and FR_POW10_FAST added as shift-only
  alternatives for 8-bit targets where 64-bit multiply is
  expensive.

New symbols

• FR_MIN, FR_MAX, FR_CLAMP — standard min/max/clamp.
• FR_DIV_TRUNC(x, xr, y, yr) — truncating division (the old
  FR_DIV behaviour).
• FR_MOD(x, xr, y, yr) — fixed-point modulus.
• Radix-28 constants: FR_kLOG2E_28, FR_krLOG2E_28,
  FR_kLOG2_10_28, FR_krLOG2_10_28.

Breaking changes from v2.0.0

Change	v2.0.0	v2.0.1
sin/cos return type	s16 (s0.15)	s32 (s15.16)
sin/cos 1.0 value	32767	65536 (exact)
tan return format	s16.15 (radix 15)	s15.16 (radix 16)
tan saturation	±(32767 << 15)	±INT32_MAX
FR_acos/asin signature	(input, radix) → s16 degrees	(input, radix, out_radix) → s32 radians
FR_atan signature	(input, radix) → s16 degrees	(input, radix, out_radix) → s32 radians
FR_atan2 signature	(y, x) → s16 degrees	(y, x, out_radix) → s32 radians
FR_BAM2RAD	off by 1024× (bug)	correct
FR_DIV rounding	truncates toward zero	rounds to nearest (use FR_DIV_TRUNC for old behaviour)

---

Version 2.0.0 (2026)

This is a significant bug-fix and precision-improvement release. Several
v1 functions produced wrong numerical results on every platform due to
arithmetic bugs; others produced wrong results specifically on 64-bit
hosts because of a typedef choice. Every changed function is covered by
the TDD characterization suite in tests/test_tdd.cpp.

See dev/fr_math_precision.md for the full per-symbol reference
(inputs, outputs, precision, saturation) and dev/fr_math2_impl_plan.md
for the implementation plan this release executed.

64-bit safety (portability)

• FR_defs.h: migrated typedefs to <stdint.h> (s8 → int8_t,
  s32 → int32_t, etc.). v1 defined s32 as signed long, which is
  64 bits on LP64 platforms (Linux and macOS on x64 / ARM64), so every
  fixed-point computation that relied on s32 being exactly 32 bits
  silently produced wrong answers on desktop Linux and macOS. C99
  <stdint.h> is now mandatory in v2 (every modern toolchain — gcc,
  clang, MSVC, IAR, Keil C51, sdcc, MSP430-gcc, AVR-gcc, RISC-V/ARM —
  ships it). If you are stuck on a pre-C99 compiler, FR_Math 1.0.x
  remains the version for you.

Numerical fixes

• FR_FixMulSat / FR_FixMuls: v1 used a split-multiply formula
  with an algebraic bug that returned wrong values for certain sign
  combinations. v2 uses an int64_t fast path with explicit
  saturation.
• FR_log2: v1 was missing the accumulator in the mantissa-table
  step and returned wrong values for non-power-of-2 inputs. v2 rewrites
  the algorithm: leading-bit-position → normalize to s1.30 → 33-entry
  mantissa lookup with linear interpolation.
• FR_ln / FR_log10: inherit the FR_log2 fix automatically
  (they are multiplies of FR_log2 by a constant).
• FR_pow2: v1 used C truncation toward zero to extract the integer
  exponent, which gave wrong answers for negative non-integer inputs
  (e.g. FR_pow2(-0.5) returned ~2 instead of ~0.707). v2 uses
  mathematical floor (toward −∞) and a 17-entry fraction table with
  linear interp.
• FR_atan2: v1 was a placeholder that returned garbage. v2 is a
  correct octant-reduced arctan with a 33-entry table. Max error ≤ 1°.
  v2 also drops the vestigial radix parameter — new signature is
  FR_atan2(s32 y, s32 x).
• FR_atan: v1 was wired up incorrectly. v2 implements as
  FR_atan2(input, 1<<radix).
• FR_Tan: v1 used s16 loop variables that overflowed for angles
  near 90°. v2 uses s32.
• FR_TanI: removed dead unreachable code (if (270 == deg)).
• FR_printNumD: v1 invoked unary minus on INT_MIN, which is
  undefined behavior, and always returned 0. v2 works in unsigned
  magnitude and returns the real byte count (or −1 on null f).
• FR_printNumF: same INT_MIN fix, plus v1's fraction extraction
  was mathematically wrong and printed bogus tail digits.
• FR_printNumH: v1 shifted a signed negative int right. v2 casts
  to unsigned first.

Macro fixes

• FR_DEG2RAD and FR_RAD2DEG: v1 had the macro bodies swapped
  relative to their names, so FR_DEG2RAD(x) actually multiplied by
  57.3 and FR_RAD2DEG(x) multiplied by 0.017. v2 swaps them back.
  FR_DEG2RAD also had missing parens around x in a subexpression.
• FR_NUM: v1 signature was FR_NUM(i, f, r) and the body ignored
  the f argument entirely. v2 adds an explicit d digit-count
  argument — new signature is FR_NUM(i, f, d, r). Any caller of the
  v1 form must be updated.

New functionality

• Radian-native trig (fr_cos, fr_sin, fr_tan, fr_cos_bam,
  fr_sin_bam, fr_cos_deg, fr_sin_deg): takes radians (or BAM, or
  integer degrees) directly rather than forcing everything through
  integer degrees. Uses a new 129-entry s0.15 quadrant cosine table in
  src/FR_trig_table.h with linear interpolation and round-to-nearest.
  Max error ≤ 1 LSB of s0.15 (~3e−5). Mean error ~0.
• BAM (Binary Angular Measure) macros: FR_DEG2BAM, FR_BAM2DEG,
  FR_RAD2BAM, FR_BAM2RAD. BAM is the natural integer representation
  for trig: 16 bits per full circle, top 2 bits select the quadrant,
  next 7 bits index the table, bottom 7 bits drive interpolation.
• Table size is a compile-time knob: -DFR_TRIG_TABLE_BITS=8 gives
  a 257-entry table for halved worst-case error (default is 7 / 129
  entries).
• Square root and hypot (FR_sqrt, FR_hypot): radix-aware fixed
  point square root and 2D vector magnitude. FR_sqrt uses a
  digit-by-digit (shift-and-subtract) integer isqrt on int64_t.
  Negative input returns FR_DOMAIN_ERROR (INT32_MIN). FR_hypot
  computes sqrt(x^2 + y^2) with no intermediate overflow up to the
  full s32 range. Bit-exact for perfect squares; max error ~1 LSB at
  the requested radix.
• Fast approximate magnitude (FR_hypot_fast, FR_hypot_fast8):
  shift-only piecewise-linear approximation of sqrt(x^2 + y^2) — no
  multiply, no divide, no 64-bit math, no ROM table, no iteration.
  FR_hypot_fast uses 4 segments (~0.4% peak error);
  FR_hypot_fast8 uses 8 segments (~0.14% peak error). Based on the
  method of US Patent 6,567,777 B1 (Chatterjee, public domain). No
  radix parameter needed — the algorithm is scale-invariant.
• Wave function family for embedded audio / LFOs / control
  signals. All take a u16 BAM phase and return s0.15 (s16, ±32767):
  • fr_wave_sqr(phase) — symmetric square wave.
  • fr_wave_pwm(phase, duty) — variabl...

FR_Math 2.0.1

FR_Math Release Notes

Version 2.0.1 (2026)

Precision and accuracy release. All changes are backward-compatible
with v2.0.0 except where noted.

Trig output precision

• fr_cos_bam / fr_sin_bam now return s15.16 (was s0.15).
  Exact values at cardinal angles: cos(0°) = 65536, cos(90°) = 0.
  FR_TRIG_ONE = 65536.
• FR_TanI / fr_tan return s15.16 (was s16.15). Saturation
  is now ±INT32_MAX.
• All wrappers updated: FR_Cos, FR_Sin, FR_CosI, FR_SinI,
  FR_TanI, FR_Tan.

Inverse trig now returns radians

• FR_acos, FR_asin, FR_atan gain an out_radix parameter
  and return radians at that radix (was degrees as s16).
• FR_atan2(y, x, out_radix) also returns radians.
• FR_BAM2RAD macro corrected (was off by a factor of 1024).

Rounding improvements

• FR_FixMuls / FR_FixMulSat: add 0.5 LSB (+0x8000) before
  the >>16 shift. Both now round to nearest instead of
  truncating.
• FR_sqrt / FR_hypot: the internal fr_isqrt64 now rounds to
  nearest (remainder > root → +1). Worst-case error drops from
  <1 LSB to ≤ 0.5 LSB.
• FR_DIV now rounds to nearest (≤ 0.5 LSB error) instead of
  truncating. FR_DIV_TRUNC preserves the old truncating behaviour.

Improved exp / log accuracy

• FR_pow2 table expanded from 17 entries (16 segments) to
  65 entries (64 segments, 260 bytes). Interpolation error drops
  by ~16×.
• FR_log2 table expanded from 33 entries to 65 entries
  (6-bit index / 24-bit interpolation). Worst-case error ≤ 4 LSB
  at Q16.16.
• FR_MULK28 macro added: multiplies any fixed-point value
  by a radix-28 constant using a 64-bit intermediate with
  round-to-nearest. ~9 decimal digits of precision.
• FR_EXP and FR_POW10 now use FR_MULK28 for the base
  conversion instead of shift-only macros.
• FR_ln and FR_log10 also use FR_MULK28 internally.
• FR_EXP_FAST and FR_POW10_FAST added as shift-only
  alternatives for 8-bit targets where 64-bit multiply is
  expensive.

New symbols

• FR_MIN, FR_MAX, FR_CLAMP — standard min/max/clamp.
• FR_DIV_TRUNC(x, xr, y, yr) — truncating division (the old
  FR_DIV behaviour).
• FR_MOD(x, xr, y, yr) — fixed-point modulus.
• Radix-28 constants: FR_kLOG2E_28, FR_krLOG2E_28,
  FR_kLOG2_10_28, FR_krLOG2_10_28.

Breaking changes from v2.0.0

Change	v2.0.0	v2.0.1
sin/cos return type	s16 (s0.15)	s32 (s15.16)
sin/cos 1.0 value	32767	65536 (exact)
tan return format	s16.15 (radix 15)	s15.16 (radix 16)
tan saturation	±(32767 << 15)	±INT32_MAX
FR_acos/asin signature	(input, radix) → s16 degrees	(input, radix, out_radix) → s32 radians
FR_atan signature	(input, radix) → s16 degrees	(input, radix, out_radix) → s32 radians
FR_atan2 signature	(y, x) → s16 degrees	(y, x, out_radix) → s32 radians
FR_BAM2RAD	off by 1024× (bug)	correct
FR_DIV rounding	truncates toward zero	rounds to nearest (use FR_DIV_TRUNC for old behaviour)

---

Version 2.0.0 (2026)

This is a significant bug-fix and precision-improvement release. Several
v1 functions produced wrong numerical results on every platform due to
arithmetic bugs; others produced wrong results specifically on 64-bit
hosts because of a typedef choice. Every changed function is covered by
the TDD characterization suite in tests/test_tdd.cpp.

See dev/fr_math_precision.md for the full per-symbol reference
(inputs, outputs, precision, saturation) and dev/fr_math2_impl_plan.md
for the implementation plan this release executed.

64-bit safety (portability)

• FR_defs.h: migrated typedefs to <stdint.h> (s8 → int8_t,
  s32 → int32_t, etc.). v1 defined s32 as signed long, which is
  64 bits on LP64 platforms (Linux and macOS on x64 / ARM64), so every
  fixed-point computation that relied on s32 being exactly 32 bits
  silently produced wrong answers on desktop Linux and macOS. C99
  <stdint.h> is now mandatory in v2 (every modern toolchain — gcc,
  clang, MSVC, IAR, Keil C51, sdcc, MSP430-gcc, AVR-gcc, RISC-V/ARM —
  ships it). If you are stuck on a pre-C99 compiler, FR_Math 1.0.x
  remains the version for you.

Numerical fixes

• FR_FixMulSat / FR_FixMuls: v1 used a split-multiply formula
  with an algebraic bug that returned wrong values for certain sign
  combinations. v2 uses an int64_t fast path with explicit
  saturation.
• FR_log2: v1 was missing the accumulator in the mantissa-table
  step and returned wrong values for non-power-of-2 inputs. v2 rewrites
  the algorithm: leading-bit-position → normalize to s1.30 → 33-entry
  mantissa lookup with linear interpolation.
• FR_ln / FR_log10: inherit the FR_log2 fix automatically
  (they are multiplies of FR_log2 by a constant).
• FR_pow2: v1 used C truncation toward zero to extract the integer
  exponent, which gave wrong answers for negative non-integer inputs
  (e.g. FR_pow2(-0.5) returned ~2 instead of ~0.707). v2 uses
  mathematical floor (toward −∞) and a 17-entry fraction table with
  linear interp.
• FR_atan2: v1 was a placeholder that returned garbage. v2 is a
  correct octant-reduced arctan with a 33-entry table. Max error ≤ 1°.
  v2 also drops the vestigial radix parameter — new signature is
  FR_atan2(s32 y, s32 x).
• FR_atan: v1 was wired up incorrectly. v2 implements as
  FR_atan2(input, 1<<radix).
• FR_Tan: v1 used s16 loop variables that overflowed for angles
  near 90°. v2 uses s32.
• FR_TanI: removed dead unreachable code (if (270 == deg)).
• FR_printNumD: v1 invoked unary minus on INT_MIN, which is
  undefined behavior, and always returned 0. v2 works in unsigned
  magnitude and returns the real byte count (or −1 on null f).
• FR_printNumF: same INT_MIN fix, plus v1's fraction extraction
  was mathematically wrong and printed bogus tail digits.
• FR_printNumH: v1 shifted a signed negative int right. v2 casts
  to unsigned first.

Macro fixes

• FR_DEG2RAD and FR_RAD2DEG: v1 had the macro bodies swapped
  relative to their names, so FR_DEG2RAD(x) actually multiplied by
  57.3 and FR_RAD2DEG(x) multiplied by 0.017. v2 swaps them back.
  FR_DEG2RAD also had missing parens around x in a subexpression.
• FR_NUM: v1 signature was FR_NUM(i, f, r) and the body ignored
  the f argument entirely. v2 adds an explicit d digit-count
  argument — new signature is FR_NUM(i, f, d, r). Any caller of the
  v1 form must be updated.

New functionality

• Radian-native trig (fr_cos, fr_sin, fr_tan, fr_cos_bam,
  fr_sin_bam, fr_cos_deg, fr_sin_deg): takes radians (or BAM, or
  integer degrees) directly rather than forcing everything through
  integer degrees. Uses a new 129-entry s0.15 quadrant cosine table in
  src/FR_trig_table.h with linear interpolation and round-to-nearest.
  Max error ≤ 1 LSB of s0.15 (~3e−5). Mean error ~0.
• BAM (Binary Angular Measure) macros: FR_DEG2BAM, FR_BAM2DEG,
  FR_RAD2BAM, FR_BAM2RAD. BAM is the natural integer representation
  for trig: 16 bits per full circle, top 2 bits select the quadrant,
  next 7 bits index the table, bottom 7 bits drive interpolation.
• Table size is a compile-time knob: -DFR_TRIG_TABLE_BITS=8 gives
  a 257-entry table for halved worst-case error (default is 7 / 129
  entries).
• Square root and hypot (FR_sqrt, FR_hypot): radix-aware fixed
  point square root and 2D vector magnitude. FR_sqrt uses a
  digit-by-digit (shift-and-subtract) integer isqrt on int64_t.
  Negative input returns FR_DOMAIN_ERROR (INT32_MIN). FR_hypot
  computes sqrt(x^2 + y^2) with no intermediate overflow up to the
  full s32 range. Bit-exact for perfect squares; max error ~1 LSB at
  the requested radix.
• Fast approximate magnitude (FR_hypot_fast, FR_hypot_fast8):
  shift-only piecewise-linear approximation of sqrt(x^2 + y^2) — no
  multiply, no divide, no 64-bit math, no ROM table, no iteration.
  FR_hypot_fast uses 4 segments (~0.4% peak error);
  FR_hypot_fast8 uses 8 segments (~0.14% peak error). Based on the
  method of US Patent 6,567,777 B1 (Chatterjee, public domain). No
  radix parameter needed — the algorithm is scale-invariant.
• Wave function family for embedded audio / LFOs / control
  signals. All take a u16 BAM phase and return s0.15 (s16, ±32767):
  • fr_wave_sqr(phase) — symmetric square wave.
  • fr_wave_pwm(phase, duty) — variable-duty pulse wave; duty is
    a u16 threshold in BAM units.
  • fr_wave_tri(phase) — symmetric triangle, peaks clamped to
    ±32767. Max error vs ideal triangle ~3e−5 (1 LSB s0.15).
  • fr_wave_saw(phase) — sawtooth, (s16)(phase - 0x8000) with
    boundary clamp.
  • fr_wave_tri_morph(phase, break_point) — variable-symmetry
    triangle that morphs into a sawtooth as break_point approaches
    0 or 0xffff. Returns unipolar [0, 32767]. Uses one division
    per sample.
  • fr_wave_noise(state) — 32-bit Galois LFSR (poly 0xD0000001,
    period 2^32 − 1) returning a full-range s16. Caller owns the
    u32 state; seed with any non-zero value.
• FR_HZ2BAM_INC(hz, sample_rate): phase increment helper.
  Computes hz * 65536 / sample_rate so a u16 phase accumulator
  driven by this increment produces the requested frequency. Example:
  FR_HZ2BAM_INC(440, 48000) = 600 (≈ 439.45 Hz).
• ADSR envelope generator (fr_adsr_t, fr_adsr_init,
  fr_adsr_trigger, fr_adsr_release, fr_adsr_step): linear-segment
  attack/decay/sustain/release envelope. Internal levels are stored as
  s1.30 so very long durations (e.g. 48000-sample attack at 48 kHz)
  still get a non-zero p...

FR_Math 2.0.0

FR_Math Release Notes

Version 2.0.0 (2026)

This is a significant bug-fix and precision-improvement release. Several
v1 functions produced wrong numerical results on every platform due to
arithmetic bugs; others produced wrong results specifically on 64-bit
hosts because of a typedef choice. Every changed function is covered by
the TDD characterization suite in tests/test_tdd.cpp.

See dev/fr_math_precision.md for the full per-symbol reference
(inputs, outputs, precision, saturation) and dev/fr_math2_impl_plan.md
for the implementation plan this release executed.

64-bit safety (portability)

• FR_defs.h: migrated typedefs to <stdint.h> (s8 → int8_t,
  s32 → int32_t, etc.). v1 defined s32 as signed long, which is
  64 bits on LP64 platforms (Linux and macOS on x64 / ARM64), so every
  fixed-point computation that relied on s32 being exactly 32 bits
  silently produced wrong answers on desktop Linux and macOS. C99
  <stdint.h> is now mandatory in v2 (every modern toolchain — gcc,
  clang, MSVC, IAR, Keil C51, sdcc, MSP430-gcc, AVR-gcc, RISC-V/ARM —
  ships it). If you are stuck on a pre-C99 compiler, FR_Math 1.0.x
  remains the version for you.

Numerical fixes

• FR_FixMulSat / FR_FixMuls: v1 used a split-multiply formula
  with an algebraic bug that returned wrong values for certain sign
  combinations. v2 uses an int64_t fast path with explicit
  saturation.
• FR_log2: v1 was missing the accumulator in the mantissa-table
  step and returned wrong values for non-power-of-2 inputs. v2 rewrites
  the algorithm: leading-bit-position → normalize to s1.30 → 33-entry
  mantissa lookup with linear interpolation.
• FR_ln / FR_log10: inherit the FR_log2 fix automatically
  (they are multiplies of FR_log2 by a constant).
• FR_pow2: v1 used C truncation toward zero to extract the integer
  exponent, which gave wrong answers for negative non-integer inputs
  (e.g. FR_pow2(-0.5) returned ~2 instead of ~0.707). v2 uses
  mathematical floor (toward −∞) and a 17-entry fraction table with
  linear interp.
• FR_atan2: v1 was a placeholder that returned garbage. v2 is a
  correct octant-reduced arctan with a 33-entry table. Max error ≤ 1°.
  v2 also drops the vestigial radix parameter — new signature is
  FR_atan2(s32 y, s32 x).
• FR_atan: v1 was wired up incorrectly. v2 implements as
  FR_atan2(input, 1<<radix).
• FR_Tan: v1 used s16 loop variables that overflowed for angles
  near 90°. v2 uses s32.
• FR_TanI: removed dead unreachable code (if (270 == deg)).
• FR_printNumD: v1 invoked unary minus on INT_MIN, which is
  undefined behavior, and always returned 0. v2 works in unsigned
  magnitude and returns the real byte count (or −1 on null f).
• FR_printNumF: same INT_MIN fix, plus v1's fraction extraction
  was mathematically wrong and printed bogus tail digits.
• FR_printNumH: v1 shifted a signed negative int right. v2 casts
  to unsigned first.

Macro fixes

• FR_DEG2RAD and FR_RAD2DEG: v1 had the macro bodies swapped
  relative to their names, so FR_DEG2RAD(x) actually multiplied by
  57.3 and FR_RAD2DEG(x) multiplied by 0.017. v2 swaps them back.
  FR_DEG2RAD also had missing parens around x in a subexpression.
• FR_NUM: v1 signature was FR_NUM(i, f, r) and the body ignored
  the f argument entirely. v2 adds an explicit d digit-count
  argument — new signature is FR_NUM(i, f, d, r). Any caller of the
  v1 form must be updated.

New functionality

• Radian-native trig (fr_cos, fr_sin, fr_tan, fr_cos_bam,
  fr_sin_bam, fr_cos_deg, fr_sin_deg): takes radians (or BAM, or
  integer degrees) directly rather than forcing everything through
  integer degrees. Uses a new 129-entry s0.15 quadrant cosine table in
  src/FR_trig_table.h with linear interpolation and round-to-nearest.
  Max error ≤ 1 LSB of s0.15 (~3e−5). Mean error ~0.
• BAM (Binary Angular Measure) macros: FR_DEG2BAM, FR_BAM2DEG,
  FR_RAD2BAM, FR_BAM2RAD. BAM is the natural integer representation
  for trig: 16 bits per full circle, top 2 bits select the quadrant,
  next 7 bits index the table, bottom 7 bits drive interpolation.
• Table size is a compile-time knob: -DFR_TRIG_TABLE_BITS=8 gives
  a 257-entry table for halved worst-case error (default is 7 / 129
  entries).
• Square root and hypot (FR_sqrt, FR_hypot): radix-aware fixed
  point square root and 2D vector magnitude. FR_sqrt uses a
  digit-by-digit (shift-and-subtract) integer isqrt on int64_t.
  Negative input returns FR_DOMAIN_ERROR (INT32_MIN). FR_hypot
  computes sqrt(x^2 + y^2) with no intermediate overflow up to the
  full s32 range. Bit-exact for perfect squares; max error ~1 LSB at
  the requested radix.
• Fast approximate magnitude (FR_hypot_fast, FR_hypot_fast8):
  shift-only piecewise-linear approximation of sqrt(x^2 + y^2) — no
  multiply, no divide, no 64-bit math, no ROM table, no iteration.
  FR_hypot_fast uses 4 segments (~0.4% peak error);
  FR_hypot_fast8 uses 8 segments (~0.14% peak error). Based on the
  method of US Patent 6,567,777 B1 (Chatterjee, public domain). No
  radix parameter needed — the algorithm is scale-invariant.
• Wave function family for embedded audio / LFOs / control
  signals. All take a u16 BAM phase and return s0.15 (s16, ±32767):
  • fr_wave_sqr(phase) — symmetric square wave.
  • fr_wave_pwm(phase, duty) — variable-duty pulse wave; duty is
    a u16 threshold in BAM units.
  • fr_wave_tri(phase) — symmetric triangle, peaks clamped to
    ±32767. Max error vs ideal triangle ~3e−5 (1 LSB s0.15).
  • fr_wave_saw(phase) — sawtooth, (s16)(phase - 0x8000) with
    boundary clamp.
  • fr_wave_tri_morph(phase, break_point) — variable-symmetry
    triangle that morphs into a sawtooth as break_point approaches
    0 or 0xffff. Returns unipolar [0, 32767]. Uses one division
    per sample.
  • fr_wave_noise(state) — 32-bit Galois LFSR (poly 0xD0000001,
    period 2^32 − 1) returning a full-range s16. Caller owns the
    u32 state; seed with any non-zero value.
• FR_HZ2BAM_INC(hz, sample_rate): phase increment helper.
  Computes hz * 65536 / sample_rate so a u16 phase accumulator
  driven by this increment produces the requested frequency. Example:
  FR_HZ2BAM_INC(440, 48000) = 600 (≈ 439.45 Hz).
• ADSR envelope generator (fr_adsr_t, fr_adsr_init,
  fr_adsr_trigger, fr_adsr_release, fr_adsr_step): linear-segment
  attack/decay/sustain/release envelope. Internal levels are stored as
  s1.30 so very long durations (e.g. 48000-sample attack at 48 kHz)
  still get a non-zero per-sample increment; fr_adsr_step returns
  s0.15 for direct multiplication into a wave sample. State machine
  exposes constants FR_ADSR_IDLE / _ATTACK / _DECAY / _SUSTAIN
  / _RELEASE.

New documentation

• dev/fr_math_precision.md — comprehensive per-symbol precision
  reference. Every public macro, constant, and function is documented
  with inputs, output format, worst-case error, saturation behavior,
  and side-effect notes.
• CONTRIBUTING.md — PR expectations, test discipline, portability
  rules, commit message format.
• tools/interp_analysis.html — interactive Chart.js analysis of
  trig interpolation methods. Compares nearest-neighbor, linear
  truncated, linear rounded, cosine interp, smoothstep, Hermite cubic,
  and Catmull-Rom on the same 129-entry table, over θ ∈ [−45°, +45°].
  Includes pan/zoom on a second chart showing the actual interpolated
  values vs Math.cos reference. Use this to verify interpolation
  claims and to pick a default for the library.
• scripts/clean_build.sh — one-shot clean of build/ and
  coverage/ directories. Handy when switching branches or debugging
  stale object files.

Breaking changes

• FR_NUM signature: FR_NUM(i, f, r) → FR_NUM(i, f, d, r). Any
  code that called the 3-argument form will fail to compile. The old
  form was broken (it ignored f entirely), so any caller was already
  getting wrong results.
• FR_atan2 signature: FR_atan2(y, x, radix) → FR_atan2(y, x).
  The radix parameter was vestigial (ignored by the v1 placeholder and
  unnecessary in v2).
• FR_RESULT and FR_E_* codes removed: v1's HRESULT-style return
  codes are gone. The matrix inv() methods now return bool
  (true on success, false if singular). add(), sub(), and
  setrotate() return void. Math functions that can hit a domain
  error return the named sentinel FR_DOMAIN_ERROR; saturating
  arithmetic returns FR_OVERFLOW_POS / FR_OVERFLOW_NEG.
• FR_SQUARE and FR_FIXMUL32u removed: these s16.16-only macros
  were narrow specializations of FR_FixMuls/FR_FixMulSat. Use the
  generic versions, which now have an int64_t fast path and work at
  any radix.
• FR_NO_INT64 and FR_NO_STDINT build flags removed: every
  C99-or-newer toolchain on every architecture (including 8-bit
  targets like sdcc, AVR-gcc, MSP430-gcc) provides <stdint.h> and
  64-bit integer arithmetic, so the conditional fallbacks were
  carrying their weight in maintenance and ROM for no benefit.
• Wider intermediate types: if you had code that poked at s32 as
  if it were long (e.g. printing with %ld), that will now warn.
  Use %d with s32 / int32_t, or cast.

Version 1.0.3 (2025)

Test Coverage Improvements

• Increased overall test coverage from 4% to 72%
  • FR_math.c: 60% coverage
  • FR_math_2D.cpp: 98% coverage
• Added comprehensive test suite with multiple test files:
  • test_comprehensive.c - Comprehensive tests for core math functions
  • test_overflow_saturation.c - Edge case an...

FR_Math v1.0.3

FR_Math Release Notes

Version 1.0.3 (2025)

Test Coverage Improvements

• Increased overall test coverage from 4% to 72%
  • FR_math.c: 60% coverage
  • FR_math_2D.cpp: 98% coverage
• Added comprehensive test suite with multiple test files:
  • test_comprehensive.c - Comprehensive tests for core math functions
  • test_overflow_saturation.c - Edge case and overflow behavior tests
  • test_full_coverage.c - Tests for previously uncovered functions
  • test_2d_complete.cpp - Complete 2D transformation matrix tests
• Fixed test failures in 2D transformation tests (XFormPtI now correctly expects integer inputs)

Bug Fixes

• Fixed FR_atan function - Was incorrectly calling FR_atan2 with wrong arguments, now properly calls FR_atanI
• Fixed constant declarations - Changed FR_PI, FR_2PI, FR_E from non-const to proper const declarations
• Fixed XFormPtI test assumptions - Tests were incorrectly passing fixed-point values instead of integers

Build System Enhancements

• Added comprehensive Makefile targets:
  • make lib - Build static library
  • make test - Run all tests
  • make coverage - Generate coverage reports with lcov
  • make examples - Build example programs
  • make clean - Clean build artifacts
• Added GitHub Actions CI/CD:
  • Multi-platform testing (Ubuntu, macOS)
  • Multi-compiler support (gcc, clang)
  • Cross-compilation testing (ARM, RISC-V)
  • 32-bit compatibility testing
  • Automated coverage reporting to Codecov
  • Overflow and saturation testing with sanitizers

Documentation Improvements

• Cleaned up README.md:
  • Fixed grammar and spelling throughout
  • Added "Building and Testing" section with clear instructions
  • Improved formatting with consistent markdown usage
  • Added supported platforms list
  • Better code examples with syntax highlighting
  • Clearer mathematical notation and explanations
• Added CLAUDE.md - Assistant-friendly documentation with:
  • Project structure overview
  • Key concepts and math notation explained
  • Testing and linting commands
  • Common pitfalls and tips
• Added inline documentation for test files explaining coverage goals

Code Quality

• Removed unused/unimplemented functions:
  • Removed FR_atan declaration (was declared but never implemented)
  • Cleaned up test code to remove references to non-existent functions
• Fixed compiler warnings:
  • Resolved deprecated C++ warnings
  • Fixed format string warnings in tests
  • Addressed unused variable warnings
• Improved code organization:
  • Better separation of test types
  • Clearer test naming conventions
  • More maintainable test structure

Platform Support

• Verified compilation and testing on:
  • x86/x64 (Linux, macOS)
  • ARM (32-bit and 64-bit)
  • RISC-V
  • 32-bit x86 targets
• Added cross-compilation support in CI

Version Updates

• Updated all source file headers to version 1.0.3
• Added version section to README.md

Previous Versions

Version 1.02

• Initial public release
• Basic fixed-point math operations
• 2D transformation matrices
• Core trigonometric functions

Version 1.01

• Internal development version
• Cleaned up naming conventions
• Initial test framework

---

Note: FR_Math has been in development since 2000, originally used in embedded systems for Palm Pilots and later ARM cores. This is the first version with comprehensive testing and CI/CD integration.