Add cached powers of 10

include/fmt/format-inl.h

@@ -280,6 +280,55 @@ const uint64_t basic_data<T>::POWERS_OF_10_64[] = {
   10000000000000000000ull
 };
 
+// Normalized 64-bit significands of pow(10, k), for k = -348, -340, ..., 340.
+// These are generated by support/compute-powers.py.
+template <typename T>
+const uint64_t basic_data<T>::POW10_SIGNIFICANDS[] = {
+  0xfa8fd5a0081c0288ull, 0xbaaee17fa23ebf76ull, 0x8b16fb203055ac76ull,
+  0xcf42894a5dce35eaull, 0x9a6bb0aa55653b2dull, 0xe61acf033d1a45dfull,
+  0xab70fe17c79ac6caull, 0xff77b1fcbebcdc4full, 0xbe5691ef416bd60cull,
+  0x8dd01fad907ffc3cull, 0xd3515c2831559a83ull, 0x9d71ac8fada6c9b5ull,
+  0xea9c227723ee8bcbull, 0xaecc49914078536dull, 0x823c12795db6ce57ull,
+  0xc21094364dfb5637ull, 0x9096ea6f3848984full, 0xd77485cb25823ac7ull,
+  0xa086cfcd97bf97f4ull, 0xef340a98172aace5ull, 0xb23867fb2a35b28eull,
+  0x84c8d4dfd2c63f3bull, 0xc5dd44271ad3cdbaull, 0x936b9fcebb25c996ull,
+  0xdbac6c247d62a584ull, 0xa3ab66580d5fdaf6ull, 0xf3e2f893dec3f126ull,
+  0xb5b5ada8aaff80b8ull, 0x87625f056c7c4a8bull, 0xc9bcff6034c13053ull,
+  0x964e858c91ba2655ull, 0xdff9772470297ebdull, 0xa6dfbd9fb8e5b88full,
+  0xf8a95fcf88747d94ull, 0xb94470938fa89bcfull, 0x8a08f0f8bf0f156bull,
+  0xcdb02555653131b6ull, 0x993fe2c6d07b7facull, 0xe45c10c42a2b3b06ull,
+  0xaa242499697392d3ull, 0xfd87b5f28300ca0eull, 0xbce5086492111aebull,
+  0x8cbccc096f5088ccull, 0xd1b71758e219652cull, 0x9c40000000000000ull,
+  0xe8d4a51000000000ull, 0xad78ebc5ac620000ull, 0x813f3978f8940984ull,
+  0xc097ce7bc90715b3ull, 0x8f7e32ce7bea5c70ull, 0xd5d238a4abe98068ull,
+  0x9f4f2726179a2245ull, 0xed63a231d4c4fb27ull, 0xb0de65388cc8ada8ull,
+  0x83c7088e1aab65dbull, 0xc45d1df942711d9aull, 0x924d692ca61be758ull,
+  0xda01ee641a708deaull, 0xa26da3999aef774aull, 0xf209787bb47d6b85ull,
+  0xb454e4a179dd1877ull, 0x865b86925b9bc5c2ull, 0xc83553c5c8965d3dull,
+  0x952ab45cfa97a0b3ull, 0xde469fbd99a05fe3ull, 0xa59bc234db398c25ull,
+  0xf6c69a72a3989f5cull, 0xb7dcbf5354e9beceull, 0x88fcf317f22241e2ull,
+  0xcc20ce9bd35c78a5ull, 0x98165af37b2153dfull, 0xe2a0b5dc971f303aull,
+  0xa8d9d1535ce3b396ull, 0xfb9b7cd9a4a7443cull, 0xbb764c4ca7a44410ull,
+  0x8bab8eefb6409c1aull, 0xd01fef10a657842cull, 0x9b10a4e5e9913129ull,
+  0xe7109bfba19c0c9dull, 0xac2820d9623bf429ull, 0x80444b5e7aa7cf85ull,
+  0xbf21e44003acdd2dull, 0x8e679c2f5e44ff8full, 0xd433179d9c8cb841ull,
+  0x9e19db92b4e31ba9ull, 0xeb96bf6ebadf77d9ull, 0xaf87023b9bf0ee6bull
+};
+
+// Binary exponents of pow(10, k), for k = -348, -340, ..., 340, corresponding
+// to significands above.
+template <typename T>
+const int16_t basic_data<T>::POW10_EXPONENTS[] = {
+  -1220, -1193, -1166, -1140, -1113, -1087, -1060, -1034, -1007,  -980,  -954,
+   -927,  -901,  -874,  -847,  -821,  -794,  -768,  -741,  -715,  -688,  -661,
+   -635,  -608,  -582,  -555,  -529,  -502,  -475,  -449,  -422,  -396,  -369,
+   -343,  -316,  -289,  -263,  -236,  -210,  -183,  -157,  -130,  -103,   -77,
+    -50,   -24,     3,    30,    56,    83,   109,   136,   162,   189,   216,
+    242,   269,   295,   322,   348,   375,   402,   428,   455,   481,   508,
+    534,   561,   588,   614,   641,   667,   694,   720,   747,   774,   800,
+    827,   853,   880,   907,   933,   960,   986,  1013,  1039,  1066
+};
+
 FMT_FUNC fp operator*(fp x, fp y) {
   // Multiply 32-bit parts of significands.
   uint64_t mask = (1ULL << 32) - 1;

include/fmt/format.h

@@ -263,10 +263,10 @@ inline fp operator-(fp x, fp y) {
 fp operator*(fp x, fp y);
 
 // Compute k such that its cached power c_k = c_k.f * pow(2, c_k.e) satisfies
-// alpha <= c_k.e + e <= alpha + 3.
+// min_exponent <= c_k.e + e <= min_exponent + 3.
-inline int compute_cached_power_index(int e, int alpha) {
+inline int compute_cached_power_index(int e, int min_exponent) {
   constexpr double one_over_log2_10 = 0.30102999566398114;  // 1 / log2(10)
-  return static_cast<int>(std::ceil((alpha - e + 63) * one_over_log2_10));
+  return static_cast<int>(std::ceil((min_exponent - e + 63) * one_over_log2_10));
 }
 
 template <typename Allocator>
@@ -795,6 +795,8 @@ template <typename T = void>
 struct FMT_API basic_data {
   static const uint32_t POWERS_OF_10_32[];
   static const uint64_t POWERS_OF_10_64[];
+  static const uint64_t POW10_SIGNIFICANDS[];
+  static const int16_t POW10_EXPONENTS[];
   static const char DIGITS[];
 };
 

support/compute-powers.py

@@ -8,9 +8,43 @@ min_exponent = -348
 max_exponent = 340
 step = 8
 significand_size = 64
-for exp in range(min_exponent, max_exponent + 1, step):
+exp_offset = 2000
-    n = 10 ** exp if exp >= 0 else 2 ** 2000 / 10 ** -exp
+
+class fp:
+    pass
+
+powers = []
+for i, exp in enumerate(range(min_exponent, max_exponent + 1, step)):
+    result = fp()
+    n = 10 ** exp if exp >= 0 else 2 ** exp_offset / 10 ** -exp
     k = significand_size + 1
     # Convert to binary and round.
-    n = (int('{:0<{}b}'.format(n, k)[:k], 2) + 1) / 2
+    binary = '{:b}'.format(n)
-    print('{:0<#16x}ull'.format(n))
+    result.f = (int('{:0<{}}'.format(binary[:k], k), 2) + 1) / 2
+    result.e = len(binary) - (exp_offset if exp < 0 else 0) - significand_size
+    powers.append(result)
+    # Sanity check.
+    exp_offset10 = 400
+    actual = result.f * 10 ** exp_offset10
+    if result.e > 0:
+        actual *= 2 ** result.e
+    else:
+        for j in range(-result.e):
+            actual /= 2
+    expected = 10 ** (exp_offset10 + exp)
+    precision = len('{}'.format(expected)) - len('{}'.format(actual - expected))
+    if precision < 19:
+        print('low precision:', precision)
+        exit(1)
+
+print('Significands:', end='')
+for i, fp in enumerate(powers):
+    if i % 3 == 0:
+        print(end='\n ')
+    print(' {:0<#16x}ull'.format(fp.f, ), end=',')
+
+print('\n\nExponents:', end='')
+for i, fp in enumerate(powers):
+    if i % 11 == 0:
+        print(end='\n ')
+    print(' {:5}'.format(fp.e), end=',')