Precision

decimal-scaled is perfectly precise — correctly rounded to within 0.5 ULP, i.e. 0 bits of error.

We execute 101,809 specialised inputs across all 28 functions, on 117 widths and scales, under all 6 rounding modes, resulting in 71,469,918 separate checks.

Each golden case was gathered from the codebase failing, from theory, from destructive testing of our own code, and from destructive testing of other libraries — collated into a behemoth of meanness.

The data on this page is generated and committed directly from the golden-comprehensive CI job and auto-rendered here; there is no manual editing. If we ever fail our guarantee, we want you to know.

Each row is a function, each column a storage width. A ✓ means 0 bits of error — the exact correctly-rounded value — across every scale and all six rounding modes at that width, with the number of checks verified correct beneath it. A ✗ marks a cell with at least one failure, the count of failing checks beneath. Inputs outside a function's domain or a tier's range aren't checks, so they count toward neither. The full per-scale, per-mode surface lives in results/golden/.

Function	D18	D38	D57	D76	D115	D153	D230	D307	D462	D616	D924	D1232
acos	✓ 14,412	✓ 75,822	✓ 106,752	✓ 109,260	✓ 118,626	✓ 96,336	✓ 120,288	✓ 216,636	✓ 168,474	✓ 192,588	✓ 192,588	✓ 192,588
acosh	✓ 34,938	✓ 104,382	✓ 109,968	✓ 110,022	✓ 113,874	✓ 88,074	✓ 113,886	✓ 217,134	✓ 165,510	✓ 191,334	✓ 191,334	✓ 191,334
add	✓ 372	✓ 1,296	✓ 1,296	✓ 1,464	✓ 1,446	✓ 1,266	✓ 1,758	✓ 3,270	✓ 2,766	✓ 3,144	✓ 3,312	✓ 3,450
asin	✓ 14,712	✓ 79,692	✓ 114,084	✓ 118,026	✓ 127,770	✓ 104,502	✓ 130,422	✓ 234,948	✓ 182,700	✓ 208,872	✓ 208,872	✓ 208,872
asinh	✓ 12,180	✓ 74,166	✓ 96,438	✓ 109,158	✓ 113,904	✓ 97,170	✓ 127,926	✓ 235,140	✓ 193,554	✓ 219,258	✓ 224,328	✓ 228,696
atan	✓ 12,558	✓ 73,050	✓ 94,356	✓ 107,178	✓ 111,450	✓ 95,262	✓ 124,980	✓ 230,082	✓ 189,138	✓ 214,242	✓ 219,090	✓ 223,236
atan2	✓ 41,946	✓ 118,128	✓ 120,768	✓ 121,494	✓ 125,406	✓ 96,768	✓ 126,162	✓ 243,096	✓ 185,262	✓ 215,112	✓ 215,688	✓ 215,976
atanh	✓ 14,448	✓ 76,842	✓ 108,990	✓ 112,818	✓ 121,602	✓ 99,474	✓ 124,212	✓ 223,734	✓ 174,006	✓ 198,936	✓ 198,960	✓ 198,960
cbrt	✓ 25,776	✓ 101,748	✓ 109,314	✓ 118,026	✓ 121,902	✓ 99,234	✓ 134,322	✓ 253,746	✓ 206,046	✓ 234,342	✓ 240,006	✓ 244,578
cos	✓ 17,514	✓ 80,178	✓ 115,536	✓ 119,958	✓ 125,100	✓ 105,168	✓ 130,770	✓ 235,806	✓ 175,890	✓ 203,400	✓ 207,054	✓ 209,586
cosh	✓ 28,428	✓ 102,522	✓ 132,384	✓ 137,202	✓ 143,838	✓ 118,614	✓ 149,430	✓ 271,944	✓ 204,012	✓ 236,472	✓ 239,298	✓ 241,068
div	✓ 246	✓ 846	✓ 828	✓ 1,008	✓ 1,050	✓ 936	✓ 1,356	✓ 2,514	✓ 2,196	✓ 2,484	✓ 2,628	✓ 2,760
exp	✓ 27,822	✓ 95,568	✓ 120,480	✓ 124,770	✓ 129,618	✓ 107,064	✓ 135,018	✓ 246,024	✓ 183,666	✓ 213,318	✓ 215,910	✓ 217,734
exp2	✓ 32,160	✓ 102,042	✓ 122,856	✓ 127,698	✓ 130,908	✓ 108,450	✓ 136,884	✓ 250,050	✓ 186,930	✓ 216,834	✓ 219,432	✓ 221,202
hypot	✓ 1,440	✓ 3,882	✓ 2,958	✓ 3,300	✓ 2,988	✓ 2,592	✓ 3,390	✓ 6,042	✓ 4,956	✓ 5,586	✓ 5,772	✓ 5,934
ln	✓ 31,182	✓ 116,082	✓ 134,544	✓ 129,720	✓ 135,840	✓ 105,696	✓ 135,960	✓ 256,920	✓ 196,536	✓ 226,902	✓ 226,998	✓ 227,052
log	✓ 14,586	✓ 46,032	✓ 44,790	✓ 45,768	✓ 44,280	✓ 34,908	✓ 44,778	✓ 84,210	✓ 64,494	✓ 74,358	✓ 74,352	✓ 74,400
log10	✓ 31,758	✓ 119,568	✓ 139,308	✓ 134,694	✓ 141,468	✓ 110,286	✓ 141,834	✓ 267,432	✓ 205,044	✓ 236,508	✓ 236,718	✓ 236,910
log2	✓ 32,412	✓ 118,218	✓ 134,340	✓ 129,468	✓ 134,874	✓ 104,958	✓ 135,138	✓ 255,444	✓ 195,570	✓ 225,738	✓ 225,918	✓ 226,032
mul	✓ 414	✓ 1,752	✓ 1,944	✓ 2,202	✓ 2,406	✓ 2,154	✓ 2,874	✓ 5,394	✓ 4,644	✓ 5,490	✓ 5,754	✓ 5,874
powf	✓ 32,436	✓ 84,846	✓ 101,244	✓ 108,024	✓ 105,042	✓ 92,472	✓ 119,196	✓ 218,358	✓ 152,184	✓ 177,240	✓ 180,222	✓ 185,424
rem	✓ 210	✓ 780	✓ 786	✓ 966	✓ 1,014	✓ 906	✓ 1,320	✓ 2,454	✓ 2,148	✓ 2,430	✓ 2,574	✓ 2,706
sin	✓ 17,544	✓ 80,286	✓ 115,968	✓ 119,988	✓ 125,400	✓ 105,246	✓ 130,998	✓ 236,124	✓ 176,100	✓ 203,634	✓ 207,474	✓ 209,898
sinh	✓ 27,984	✓ 97,974	✓ 125,184	✓ 129,870	✓ 135,252	✓ 111,882	✓ 141,054	✓ 256,680	✓ 192,072	✓ 222,906	✓ 225,894	✓ 227,598
sqrt	✓ 27,390	✓ 104,052	✓ 109,236	✓ 117,906	✓ 121,650	✓ 98,514	✓ 133,182	✓ 251,574	✓ 204,144	✓ 232,134	✓ 237,330	✓ 241,698
sub	✓ 372	✓ 1,296	✓ 1,296	✓ 1,464	✓ 1,446	✓ 1,266	✓ 1,758	✓ 3,270	✓ 2,766	✓ 3,144	✓ 3,312	✓ 3,450
tan	✓ 17,274	✓ 79,482	✓ 114,594	✓ 120,888	✓ 125,490	✓ 106,128	✓ 131,178	✓ 237,192	✓ 177,090	✓ 204,672	✓ 208,266	✓ 210,750
tanh	✓ 17,208	✓ 83,532	✓ 118,536	✓ 121,524	✓ 131,232	✓ 106,866	✓ 133,710	✓ 241,296	✓ 183,060	✓ 212,682	✓ 215,142	✓ 215,466