ONE SIGHT, THREE TRADITIONS

The classic haversine reduction: hav ZD = hav(lat − dec) + cos lat · cos dec · hav LHA, then Hc = 90° − ZD. The multiplication is the expensive part — so the navigator never multiplies. The log table turns the product into a sum of five-digit integers: look up each logarithm, add the mantissas in columns, take the antilog. The decimal point lives in the navigator's head; the paper only ever carries integers.

Recomputed here to five places with the same rules a navigator followed (not scanned from a historical volume). Azimuth went the same way — time-azimuth tables or ABC tables: more lookups, more integer addition. Table precision, typically 0.1′, was a declared quantization step: everyone knew exactly how coarse the tools were.

A calculator or computer evaluates sin Hc = sin φ sin δ + cos φ cos δ cos LHA directly in double precision: Hc = 66.680325…°, with ~16 significant digits of arithmetic headroom. Precision is not the issue — a double is superb at this. Two things are quietly different from the tables, though. First, the scale stops being shared: whole degrees like 40°, 20°, 346° happen to be exactly representable, but rounding begins the moment they are converted to radians, and a typical fractional reading like 25°14.3′ is already the nearest binary fraction, not the value itself (page 2 makes this concrete). Second, the last bits are not fixed by the C/libm contract: libm is a library choice, and two correct math libraries may legitimately round a sine differently, so identical inputs can differ in the final bit across platforms. Invisible for navigation accuracy — but identical results are not guaranteed unless an application deliberately engineers and pins them.

The shipped binary, same sight (angles as integer centidegrees, 100 = 1°; the trailing argument is a corrected altitude Ho in integer milli-arcminutes, from which the intercept follows):

$ ./sight_reduction --reduce 4000 -7400 6000 2000 4012800
Hc(A): 66.68 deg
Hc(B): 66.68 deg
Hc(C): 66.68 deg   (machine sin_hc=986027972/2^30)
Zn(A): 144.95 deg true
Zn(B): 144.96 deg true   square-key=27022/65536
Zn(C): 144.96 deg true   square-key=27022/65536
A/B altitude difference: 0.000'
Intercept: 12.0 nm TOWARD

Same Hc as the log tables, to their own precision — and the machine's trick is the same kind of trick. The log table replaced the human's multiplication with table lookup and integer addition. CORDIC replaces the machine's trigonometry with table lookup and integer shift-and-add: rotate a vector through a fixed ladder of angles arctan(2⁻ⁱ), and each step uses one angle-table entry, two shifts, and additions or subtractions — basic operations available even on small integer CPUs. Watch the ladder home in on this very sight's azimuth:

The native-double reference azimuth of this sight is about 144.952°. In the transcript above, method A prints 144.95 while B and C print 144.96 — adjacent cells of the declared 0.01° output grid, reached by different deterministic integer routes. That pair is pinned by the golden schedule across the tested host compiler/backend matrix; the embedded profiles separately pin their own scheduled outputs.

The same ladder in the machine's own digits — a pedagogical integer trace (rotation mode, computing sin 45° = cos 45°; shown in Q2.30-style values with micro-degree residuals for readability — the shipped library's actual state is Q16.48 over 48 iterations, against an angle table it builds once at init by integer series):

Every cell is an integer. σ is the rotation direction (sign of z); there is no multiplication anywhere in the loop — only shifts, adds, and the angle table (48 entries in the shipped library). The same ladder run in vectoring mode gives atan2 and asin, which is how Zn comes out.

And one step further: Method C drops angles out of the hot path entirely. If the almanac publishes each body as an integer unit vector (Q2.30 — 30 fractional bits), the whole sight collapses to three dot products and a divide. Vega, from the built-in vector almanac:

$ ./sight_reduction --star 3 836136000000
Vega at UT1 J2000 +836136000000 ms:
GHA Aries: 279.06 deg
GHA:       359.61 deg
dec:       38.81 deg
earth-fixed vector: (836698133, 5757400, 672922426)/2^30

sin Hc = O · B as one integer dot product: the measured hot-path cost on a laptop is ~7 ns per sight against ~32 ns for the double/libm angle path. The 7 ns figure starts from prebuilt vectors and returns machine-native outputs; almanac construction and conversion back to human angles are deliberately outside that comparison. Page 3 tours the whole chain.