optimizing d2string() and addReplyDouble() with grisu2: double to string conversion based on Florian Loitsch's Grisu-algorithm (#10587)

All commands / use cases that heavily rely on double to a string representation conversion,
(e.g. meaning take a double-precision floating-point number like 1.5 and return a string like "1.5" ),
could benefit from a performance boost by swapping snprintf(buf,len,"%.17g",value) by the
equivalent [fpconv_dtoa](https://github.com/night-shift/fpconv) or any other algorithm that ensures
100% coverage of conversion.

This is a well-studied topic and Projects like MongoDB. RedPanda, PyTorch leverage libraries
( fmtlib ) that use the optimized double to string conversion underneath.


The positive impact can be substantial. This PR uses the grisu2 approach ( grisu explained on
https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf section 5 ). 

test suite changes:
Despite being compatible, in some cases it produces a different result from printf, and some tests
had to be adjusted.
one case is that `%.17g` (which means %e or %f which ever is shorter), chose to use `5000000000`
instead of 5e+9, which sounds like a bug?
In other cases, we changed TCL to compare numbers instead of strings to ignore minor rounding
issues (`expr 0.8 == 0.79999999999999999`) 

7 files changed, 617 insertions, 0 deletions

diff --git a/deps/Makefile b/deps/Makefile
index 96dbb8c1d..c0dc71120 100644
--- a/deps/Makefile
+++ b/deps/Makefile
@@ -40,6 +40,7 @@ distclean:
 	-(cd lua && $(MAKE) clean) > /dev/null || true
 	-(cd jemalloc && [ -f Makefile ] && $(MAKE) distclean) > /dev/null || true
 	-(cd hdr_histogram && $(MAKE) clean) > /dev/null || true
+	-(cd fpconv && $(MAKE) clean) > /dev/null || true
 	-(rm -f .make-*)
 
 .PHONY: distclean
@@ -66,6 +67,12 @@ hdr_histogram: .make-prerequisites
 
 .PHONY: hdr_histogram
 
+fpconv: .make-prerequisites
+	@printf '%b %b\n' $(MAKECOLOR)MAKE$(ENDCOLOR) $(BINCOLOR)$@$(ENDCOLOR)
+	cd fpconv && $(MAKE)
+
+.PHONY: fpconv
+
 ifeq ($(uname_S),SunOS)
 	# Make isinf() available
 	LUA_CFLAGS= -D__C99FEATURES__=1
diff --git a/deps/fpconv/LICENSE.txt b/deps/fpconv/LICENSE.txt
new file mode 100644
index 000000000..36b7cd93c
--- /dev/null
+++ b/deps/fpconv/LICENSE.txt
@@ -0,0 +1,23 @@
+Boost Software License - Version 1.0 - August 17th, 2003
+
+Permission is hereby granted, free of charge, to any person or organization
+obtaining a copy of the software and accompanying documentation covered by
+this license (the "Software") to use, reproduce, display, distribute,
+execute, and transmit the Software, and to prepare derivative works of the
+Software, and to permit third-parties to whom the Software is furnished to
+do so, all subject to the following:
+
+The copyright notices in the Software and this entire statement, including
+the above license grant, this restriction and the following disclaimer,
+must be included in all copies of the Software, in whole or in part, and
+all derivative works of the Software, unless such copies or derivative
+works are solely in the form of machine-executable object code generated by
+a source language processor.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
+SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
+FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
+ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+DEALINGS IN THE SOFTWARE.
diff --git a/deps/fpconv/Makefile b/deps/fpconv/Makefile
new file mode 100644
index 000000000..2654888e4
--- /dev/null
+++ b/deps/fpconv/Makefile
@@ -0,0 +1,27 @@
+STD=
+WARN= -Wall
+OPT= -Os
+
+R_CFLAGS= $(STD) $(WARN) $(OPT) $(DEBUG) $(CFLAGS)
+R_LDFLAGS= $(LDFLAGS)
+DEBUG= -g
+
+R_CC=$(CC) $(R_CFLAGS)
+R_LD=$(CC) $(R_LDFLAGS)
+
+AR= ar
+ARFLAGS= rcs
+
+libfpconv.a: fpconv_dtoa.o
+	$(AR) $(ARFLAGS) $@ $+
+
+fpconv_dtoa.o: fpconv_dtoa.h fpconv_dtoa.c
+
+.c.o:
+	$(R_CC) -c  $< 
+
+clean:
+	rm -f *.o
+	rm -f *.a
+
+
diff --git a/deps/fpconv/README.md b/deps/fpconv/README.md
new file mode 100644
index 000000000..d64ca685a
--- /dev/null
+++ b/deps/fpconv/README.md
@@ -0,0 +1,9 @@
+libfpconv
+
+----------------------------------------------
+
+Fast and accurate double to string conversion based on Florian Loitsch's Grisu-algorithm[1].
+
+This port contains a subset of the 'C' version of Fast and accurate double to string conversion based on Florian Loitsch's Grisu-algorithm available at [github.com/night-shift/fpconv](https://github.com/night-shift/fpconv)).
+
+[1] https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf
diff --git a/deps/fpconv/fpconv_dtoa.c b/deps/fpconv/fpconv_dtoa.c
new file mode 100644
index 000000000..ad2f2de74
--- /dev/null
+++ b/deps/fpconv/fpconv_dtoa.c
@@ -0,0 +1,373 @@
+/* fpconv_dtoa.c -- floating point conversion utilities.
+ *
+ * Fast and accurate double to string conversion based on Florian Loitsch's
+ * Grisu-algorithm[1].
+ *
+ * [1] https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf
+ * ----------------------------------------------------------------------------
+ *
+ * Copyright (c) 2013-2019, night-shift <as.smljk at gmail dot com>
+ * Copyright (c) 2009, Florian Loitsch < florian.loitsch at inria dot fr >
+ * All rights reserved.
+ *
+ * Boost Software License - Version 1.0 - August 17th, 2003
+ *
+ * Permission is hereby granted, free of charge, to any person or organization
+ * obtaining a copy of the software and accompanying documentation covered by
+ * this license (the "Software") to use, reproduce, display, distribute,
+ * execute, and transmit the Software, and to prepare derivative works of the
+ * Software, and to permit third-parties to whom the Software is furnished to
+ * do so, all subject to the following:
+ *
+ * The copyright notices in the Software and this entire statement, including
+ * the above license grant, this restriction and the following disclaimer,
+ * must be included in all copies of the Software, in whole or in part, and
+ * all derivative works of the Software, unless such copies or derivative
+ * works are solely in the form of machine-executable object code generated by
+ * a source language processor.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+ * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+ * FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
+ * SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
+ * FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
+ * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+ * DEALINGS IN THE SOFTWARE.
+ */
+
+#include "fpconv_dtoa.h"
+
+#include "fpconv_powers.h"
+
+#include <stdbool.h>
+#include <string.h>
+
+#define fracmask 0x000FFFFFFFFFFFFFU
+#define expmask 0x7FF0000000000000U
+#define hiddenbit 0x0010000000000000U
+#define signmask 0x8000000000000000U
+#define expbias (1023 + 52)
+
+#define absv(n) ((n) < 0 ? -(n) : (n))
+#define minv(a, b) ((a) < (b) ? (a) : (b))
+
+static uint64_t tens[] = { 10000000000000000000U,
+                           1000000000000000000U,
+                           100000000000000000U,
+                           10000000000000000U,
+                           1000000000000000U,
+                           100000000000000U,
+                           10000000000000U,
+                           1000000000000U,
+                           100000000000U,
+                           10000000000U,
+                           1000000000U,
+                           100000000U,
+                           10000000U,
+                           1000000U,
+                           100000U,
+                           10000U,
+                           1000U,
+                           100U,
+                           10U,
+                           1U };
+
+static inline uint64_t get_dbits(double d) {
+    union
+    {
+        double dbl;
+        uint64_t i;
+    } dbl_bits = { d };
+
+    return dbl_bits.i;
+}
+
+static Fp build_fp(double d) {
+    uint64_t bits = get_dbits(d);
+
+    Fp fp;
+    fp.frac = bits & fracmask;
+    fp.exp = (bits & expmask) >> 52;
+
+    if (fp.exp) {
+        fp.frac += hiddenbit;
+        fp.exp -= expbias;
+
+    } else {
+        fp.exp = -expbias + 1;
+    }
+
+    return fp;
+}
+
+static void normalize(Fp *fp) {
+    while ((fp->frac & hiddenbit) == 0) {
+        fp->frac <<= 1;
+        fp->exp--;
+    }
+
+    int shift = 64 - 52 - 1;
+    fp->frac <<= shift;
+    fp->exp -= shift;
+}
+
+static void get_normalized_boundaries(Fp *fp, Fp *lower, Fp *upper) {
+    upper->frac = (fp->frac << 1) + 1;
+    upper->exp = fp->exp - 1;
+
+    while ((upper->frac & (hiddenbit << 1)) == 0) {
+        upper->frac <<= 1;
+        upper->exp--;
+    }
+
+    int u_shift = 64 - 52 - 2;
+
+    upper->frac <<= u_shift;
+    upper->exp = upper->exp - u_shift;
+
+    int l_shift = fp->frac == hiddenbit ? 2 : 1;
+
+    lower->frac = (fp->frac << l_shift) - 1;
+    lower->exp = fp->exp - l_shift;
+
+    lower->frac <<= lower->exp - upper->exp;
+    lower->exp = upper->exp;
+}
+
+static Fp multiply(Fp *a, Fp *b) {
+    const uint64_t lomask = 0x00000000FFFFFFFF;
+
+    uint64_t ah_bl = (a->frac >> 32) * (b->frac & lomask);
+    uint64_t al_bh = (a->frac & lomask) * (b->frac >> 32);
+    uint64_t al_bl = (a->frac & lomask) * (b->frac & lomask);
+    uint64_t ah_bh = (a->frac >> 32) * (b->frac >> 32);
+
+    uint64_t tmp = (ah_bl & lomask) + (al_bh & lomask) + (al_bl >> 32);
+    /* round up */
+    tmp += 1U << 31;
+
+    Fp fp = { ah_bh + (ah_bl >> 32) + (al_bh >> 32) + (tmp >> 32), a->exp + b->exp + 64 };
+
+    return fp;
+}
+
+static void round_digit(char *digits,
+                        int ndigits,
+                        uint64_t delta,
+                        uint64_t rem,
+                        uint64_t kappa,
+                        uint64_t frac) {
+    while (rem < frac && delta - rem >= kappa &&
+           (rem + kappa < frac || frac - rem > rem + kappa - frac)) {
+        digits[ndigits - 1]--;
+        rem += kappa;
+    }
+}
+
+static int generate_digits(Fp *fp, Fp *upper, Fp *lower, char *digits, int *K) {
+    uint64_t wfrac = upper->frac - fp->frac;
+    uint64_t delta = upper->frac - lower->frac;
+
+    Fp one;
+    one.frac = 1ULL << -upper->exp;
+    one.exp = upper->exp;
+
+    uint64_t part1 = upper->frac >> -one.exp;
+    uint64_t part2 = upper->frac & (one.frac - 1);
+
+    int idx = 0, kappa = 10;
+    uint64_t *divp;
+    /* 1000000000 */
+    for (divp = tens + 10; kappa > 0; divp++) {
+        uint64_t div = *divp;
+        unsigned digit = part1 / div;
+
+        if (digit || idx) {
+            digits[idx++] = digit + '0';
+        }
+
+        part1 -= digit * div;
+        kappa--;
+
+        uint64_t tmp = (part1 << -one.exp) + part2;
+        if (tmp <= delta) {
+            *K += kappa;
+            round_digit(digits, idx, delta, tmp, div << -one.exp, wfrac);
+
+            return idx;
+        }
+    }
+
+    /* 10 */
+    uint64_t *unit = tens + 18;
+
+    while (true) {
+        part2 *= 10;
+        delta *= 10;
+        kappa--;
+
+        unsigned digit = part2 >> -one.exp;
+        if (digit || idx) {
+            digits[idx++] = digit + '0';
+        }
+
+        part2 &= one.frac - 1;
+        if (part2 < delta) {
+            *K += kappa;
+            round_digit(digits, idx, delta, part2, one.frac, wfrac * *unit);
+
+            return idx;
+        }
+
+        unit--;
+    }
+}
+
+static int grisu2(double d, char *digits, int *K) {
+    Fp w = build_fp(d);
+
+    Fp lower, upper;
+    get_normalized_boundaries(&w, &lower, &upper);
+
+    normalize(&w);
+
+    int k;
+    Fp cp = find_cachedpow10(upper.exp, &k);
+
+    w = multiply(&w, &cp);
+    upper = multiply(&upper, &cp);
+    lower = multiply(&lower, &cp);
+
+    lower.frac++;
+    upper.frac--;
+
+    *K = -k;
+
+    return generate_digits(&w, &upper, &lower, digits, K);
+}
+
+static int emit_digits(char *digits, int ndigits, char *dest, int K, bool neg) {
+    int exp = absv(K + ndigits - 1);
+
+    /* write plain integer */
+    if (K >= 0 && (exp < (ndigits + 7))) {
+        memcpy(dest, digits, ndigits);
+        memset(dest + ndigits, '0', K);
+
+        return ndigits + K;
+    }
+
+    /* write decimal w/o scientific notation */
+    if (K < 0 && (K > -7 || exp < 4)) {
+        int offset = ndigits - absv(K);
+        /* fp < 1.0 -> write leading zero */
+        if (offset <= 0) {
+            offset = -offset;
+            dest[0] = '0';
+            dest[1] = '.';
+            memset(dest + 2, '0', offset);
+            memcpy(dest + offset + 2, digits, ndigits);
+
+            return ndigits + 2 + offset;
+
+            /* fp > 1.0 */
+        } else {
+            memcpy(dest, digits, offset);
+            dest[offset] = '.';
+            memcpy(dest + offset + 1, digits + offset, ndigits - offset);
+
+            return ndigits + 1;
+        }
+    }
+
+    /* write decimal w/ scientific notation */
+    ndigits = minv(ndigits, 18 - neg);
+
+    int idx = 0;
+    dest[idx++] = digits[0];
+
+    if (ndigits > 1) {
+        dest[idx++] = '.';
+        memcpy(dest + idx, digits + 1, ndigits - 1);
+        idx += ndigits - 1;
+    }
+
+    dest[idx++] = 'e';
+
+    char sign = K + ndigits - 1 < 0 ? '-' : '+';
+    dest[idx++] = sign;
+
+    int cent = 0;
+
+    if (exp > 99) {
+        cent = exp / 100;
+        dest[idx++] = cent + '0';
+        exp -= cent * 100;
+    }
+    if (exp > 9) {
+        int dec = exp / 10;
+        dest[idx++] = dec + '0';
+        exp -= dec * 10;
+
+    } else if (cent) {
+        dest[idx++] = '0';
+    }
+
+    dest[idx++] = exp % 10 + '0';
+
+    return idx;
+}
+
+static int filter_special(double fp, char *dest) {
+    if (fp == 0.0) {
+        dest[0] = '0';
+        return 1;
+    }
+
+    uint64_t bits = get_dbits(fp);
+
+    bool nan = (bits & expmask) == expmask;
+
+    if (!nan) {
+        return 0;
+    }
+
+    if (bits & fracmask) {
+        dest[0] = 'n';
+        dest[1] = 'a';
+        dest[2] = 'n';
+
+    } else {
+        dest[0] = 'i';
+        dest[1] = 'n';
+        dest[2] = 'f';
+    }
+
+    return 3;
+}
+
+int fpconv_dtoa(double d, char dest[24]) {
+    char digits[18];
+
+    int str_len = 0;
+    bool neg = false;
+
+    if (get_dbits(d) & signmask) {
+        dest[0] = '-';
+        str_len++;
+        neg = true;
+    }
+
+    int spec = filter_special(d, dest + str_len);
+
+    if (spec) {
+        return str_len + spec;
+    }
+
+    int K = 0;
+    int ndigits = grisu2(d, digits, &K);
+
+    str_len += emit_digits(digits, ndigits, dest + str_len, K, neg);
+
+    return str_len;
+}
diff --git a/deps/fpconv/fpconv_dtoa.h b/deps/fpconv/fpconv_dtoa.h
new file mode 100644
index 000000000..328ed83f0
--- /dev/null
+++ b/deps/fpconv/fpconv_dtoa.h
@@ -0,0 +1,45 @@
+/* fpconv_dtoa.h -- floating point conversion utilities.
+ *
+ * Fast and accurate double to string conversion based on Florian Loitsch's
+ * Grisu-algorithm[1].
+ *
+ * [1] https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf
+ * ----------------------------------------------------------------------------
+ *
+ * Copyright (c) 2013-2019, night-shift <as.smljk at gmail dot com>
+ * Copyright (c) 2009, Florian Loitsch < florian.loitsch at inria dot fr >
+ * All rights reserved.
+ *
+ * Boost Software License - Version 1.0 - August 17th, 2003
+ *
+ * Permission is hereby granted, free of charge, to any person or organization
+ * obtaining a copy of the software and accompanying documentation covered by
+ * this license (the "Software") to use, reproduce, display, distribute,
+ * execute, and transmit the Software, and to prepare derivative works of the
+ * Software, and to permit third-parties to whom the Software is furnished to
+ * do so, all subject to the following:
+ *
+ * The copyright notices in the Software and this entire statement, including
+ * the above license grant, this restriction and the following disclaimer,
+ * must be included in all copies of the Software, in whole or in part, and
+ * all derivative works of the Software, unless such copies or derivative
+ * works are solely in the form of machine-executable object code generated by
+ * a source language processor.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+ * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+ * FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
+ * SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
+ * FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
+ * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+ * DEALINGS IN THE SOFTWARE.
+ */
+
+#ifndef FPCONV_DTOA_H
+#define FPCONV_DTOA_H
+
+int fpconv_dtoa(double fp, char dest[24]);
+
+#endif
+
+/* [1] http://florian.loitsch.com/publications/dtoa-pldi2010.pdf */
diff --git a/deps/fpconv/fpconv_powers.h b/deps/fpconv/fpconv_powers.h
new file mode 100644
index 000000000..bc488f6dd
--- /dev/null
+++ b/deps/fpconv/fpconv_powers.h
@@ -0,0 +1,133 @@
+/* fpconv_powers.h -- floating point conversion utilities.
+ *
+ * Fast and accurate double to string conversion based on Florian Loitsch's
+ * Grisu-algorithm[1].
+ *
+ * [1] https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf
+ * ----------------------------------------------------------------------------
+ *
+ * Copyright (c) 2021, Redis Labs
+ * Copyright (c) 2013-2019, night-shift <as.smljk at gmail dot com>
+ * Copyright (c) 2009, Florian Loitsch < florian.loitsch at inria dot fr >
+ * All rights reserved.
+ *
+ * Boost Software License - Version 1.0 - August 17th, 2003
+ *
+ * Permission is hereby granted, free of charge, to any person or organization
+ * obtaining a copy of the software and accompanying documentation covered by
+ * this license (the "Software") to use, reproduce, display, distribute,
+ * execute, and transmit the Software, and to prepare derivative works of the
+ * Software, and to permit third-parties to whom the Software is furnished to
+ * do so, all subject to the following:
+ *
+ * The copyright notices in the Software and this entire statement, including
+ * the above license grant, this restriction and the following disclaimer,
+ * must be included in all copies of the Software, in whole or in part, and
+ * all derivative works of the Software, unless such copies or derivative
+ * works are solely in the form of machine-executable object code generated by
+ * a source language processor.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+ * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+ * FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
+ * SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
+ * FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE,
+ * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
+ * DEALINGS IN THE SOFTWARE.
+ */
+
+#include <stdint.h>
+
+#define npowers     87
+#define steppowers  8
+#define firstpower -348 /* 10 ^ -348 */
+
+#define expmax     -32
+#define expmin     -60
+
+
+typedef struct Fp {
+    uint64_t frac;
+    int exp;
+} Fp;
+
+static Fp powers_ten[] = {
+    { 18054884314459144840U, -1220 }, { 13451937075301367670U, -1193 },
+    { 10022474136428063862U, -1166 }, { 14934650266808366570U, -1140 },
+    { 11127181549972568877U, -1113 }, { 16580792590934885855U, -1087 },
+    { 12353653155963782858U, -1060 }, { 18408377700990114895U, -1034 },
+    { 13715310171984221708U, -1007 }, { 10218702384817765436U, -980 },
+    { 15227053142812498563U, -954 },  { 11345038669416679861U, -927 },
+    { 16905424996341287883U, -901 },  { 12595523146049147757U, -874 },
+    { 9384396036005875287U,  -847 },  { 13983839803942852151U, -821 },
+    { 10418772551374772303U, -794 },  { 15525180923007089351U, -768 },
+    { 11567161174868858868U, -741 },  { 17236413322193710309U, -715 },
+    { 12842128665889583758U, -688 },  { 9568131466127621947U,  -661 },
+    { 14257626930069360058U, -635 },  { 10622759856335341974U, -608 },
+    { 15829145694278690180U, -582 },  { 11793632577567316726U, -555 },
+    { 17573882009934360870U, -529 },  { 13093562431584567480U, -502 },
+    { 9755464219737475723U,  -475 },  { 14536774485912137811U, -449 },
+    { 10830740992659433045U, -422 },  { 16139061738043178685U, -396 },
+    { 12024538023802026127U, -369 },  { 17917957937422433684U, -343 },
+    { 13349918974505688015U, -316 },  { 9946464728195732843U,  -289 },
+    { 14821387422376473014U, -263 },  { 11042794154864902060U, -236 },
+    { 16455045573212060422U, -210 },  { 12259964326927110867U, -183 },
+    { 18268770466636286478U, -157 },  { 13611294676837538539U, -130 },
+    { 10141204801825835212U, -103 },  { 15111572745182864684U, -77 },
+    { 11258999068426240000U, -50 },   { 16777216000000000000U, -24 },
+    { 12500000000000000000U,   3 },   { 9313225746154785156U,   30 },
+    { 13877787807814456755U,  56 },   { 10339757656912845936U,  83 },
+    { 15407439555097886824U, 109 },   { 11479437019748901445U, 136 },
+    { 17105694144590052135U, 162 },   { 12744735289059618216U, 189 },
+    { 9495567745759798747U,  216 },   { 14149498560666738074U, 242 },
+    { 10542197943230523224U, 269 },   { 15709099088952724970U, 295 },
+    { 11704190886730495818U, 322 },   { 17440603504673385349U, 348 },
+    { 12994262207056124023U, 375 },   { 9681479787123295682U,  402 },
+    { 14426529090290212157U, 428 },   { 10748601772107342003U, 455 },
+    { 16016664761464807395U, 481 },   { 11933345169920330789U, 508 },
+    { 17782069995880619868U, 534 },   { 13248674568444952270U, 561 },
+    { 9871031767461413346U,  588 },   { 14708983551653345445U, 614 },
+    { 10959046745042015199U, 641 },   { 16330252207878254650U, 667 },
+    { 12166986024289022870U, 694 },   { 18130221999122236476U, 720 },
+    { 13508068024458167312U, 747 },   { 10064294952495520794U, 774 },
+    { 14996968138956309548U, 800 },   { 11173611982879273257U, 827 },
+    { 16649979327439178909U, 853 },   { 12405201291620119593U, 880 },
+    { 9242595204427927429U,  907 },   { 13772540099066387757U, 933 },
+    { 10261342003245940623U, 960 },   { 15290591125556738113U, 986 },
+    { 11392378155556871081U, 1013 },  { 16975966327722178521U, 1039 },
+    { 12648080533535911531U, 1066 }
+};
+
+/**
+ *  Grisu needs a cache of precomputed powers-of-ten.
+ *  This function, given an exponent and an integer k
+ *  return the normalized floating-point approximation of the power of 10.
+ * @param exp
+ * @param k
+ * @return
+ */
+static Fp find_cachedpow10(int exp, int* k)
+{
+    const double one_log_ten = 0.30102999566398114;
+
+    const int approx = -(exp + npowers) * one_log_ten;
+    int idx = (approx - firstpower) / steppowers;
+
+    while(1) {
+        int current = exp + powers_ten[idx].exp + 64;
+
+        if(current < expmin) {
+            idx++;
+            continue;
+        }
+
+        if(current > expmax) {
+            idx--;
+            continue;
+        }
+
+        *k = (firstpower + idx * steppowers);
+
+        return powers_ten[idx];
+    }
+}