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sortix--sortix/libm/src/s_exp2.c
Jonas 'Sortie' Termansen 5980be9b3c Add Sortix Math Library. 
This work is based in part on code from NetBSD libm, libc and kernel.

The library is partly public domain and partly BSD-style licensed.
2013-12-17 14:30:39 +01:00

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/*-
* Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
*/
#include <sys/cdefs.h>
__RCSID("$NetBSD: s_exp2.c,v 1.2 2010/01/11 23:38:24 christos Exp $");
#ifdef __FBSDID
__FBSDID("$FreeBSD: src/lib/msun/src/s_exp2.c,v 1.7 2008/02/22 02:27:34 das Exp $");
#endif
#include <float.h>
#include "math.h"
#include "math_private.h"
#define TBLBITS 8
#define TBLSIZE (1 << TBLBITS)
static const double
huge = 0x1p1000,
redux = 0x1.8p52 / TBLSIZE,
P1 = 0x1.62e42fefa39efp-1,
P2 = 0x1.ebfbdff82c575p-3,
P3 = 0x1.c6b08d704a0a6p-5,
P4 = 0x1.3b2ab88f70400p-7,
P5 = 0x1.5d88003875c74p-10;
static volatile double twom1000 = 0x1p-1000;
static const double tbl[TBLSIZE * 2] = {
/* exp2(z + eps) eps */
0x1.6a09e667f3d5dp-1, 0x1.9880p-44,
0x1.6b052fa751744p-1, 0x1.8000p-50,
0x1.6c012750bd9fep-1, -0x1.8780p-45,
0x1.6cfdcddd476bfp-1, 0x1.ec00p-46,
0x1.6dfb23c651a29p-1, -0x1.8000p-50,
0x1.6ef9298593ae3p-1, -0x1.c000p-52,
0x1.6ff7df9519386p-1, -0x1.fd80p-45,
0x1.70f7466f42da3p-1, -0x1.c880p-45,
0x1.71f75e8ec5fc3p-1, 0x1.3c00p-46,
0x1.72f8286eacf05p-1, -0x1.8300p-44,
0x1.73f9a48a58152p-1, -0x1.0c00p-47,
0x1.74fbd35d7ccfcp-1, 0x1.f880p-45,
0x1.75feb564267f1p-1, 0x1.3e00p-47,
0x1.77024b1ab6d48p-1, -0x1.7d00p-45,
0x1.780694fde5d38p-1, -0x1.d000p-50,
0x1.790b938ac1d00p-1, 0x1.3000p-49,
0x1.7a11473eb0178p-1, -0x1.d000p-49,
0x1.7b17b0976d060p-1, 0x1.0400p-45,
0x1.7c1ed0130c133p-1, 0x1.0000p-53,
0x1.7d26a62ff8636p-1, -0x1.6900p-45,
0x1.7e2f336cf4e3bp-1, -0x1.2e00p-47,
0x1.7f3878491c3e8p-1, -0x1.4580p-45,
0x1.80427543e1b4ep-1, 0x1.3000p-44,
0x1.814d2add1071ap-1, 0x1.f000p-47,
0x1.82589994ccd7ep-1, -0x1.1c00p-45,
0x1.8364c1eb942d0p-1, 0x1.9d00p-45,
0x1.8471a4623cab5p-1, 0x1.7100p-43,
0x1.857f4179f5bbcp-1, 0x1.2600p-45,
0x1.868d99b4491afp-1, -0x1.2c40p-44,
0x1.879cad931a395p-1, -0x1.3000p-45,
0x1.88ac7d98a65b8p-1, -0x1.a800p-45,
0x1.89bd0a4785800p-1, -0x1.d000p-49,
0x1.8ace5422aa223p-1, 0x1.3280p-44,
0x1.8be05bad619fap-1, 0x1.2b40p-43,
0x1.8cf3216b54383p-1, -0x1.ed00p-45,
0x1.8e06a5e08664cp-1, -0x1.0500p-45,
0x1.8f1ae99157807p-1, 0x1.8280p-45,
0x1.902fed0282c0ep-1, -0x1.cb00p-46,
0x1.9145b0b91ff96p-1, -0x1.5e00p-47,
0x1.925c353aa2ff9p-1, 0x1.5400p-48,
0x1.93737b0cdc64ap-1, 0x1.7200p-46,
0x1.948b82b5f98aep-1, -0x1.9000p-47,
0x1.95a44cbc852cbp-1, 0x1.5680p-45,
0x1.96bdd9a766f21p-1, -0x1.6d00p-44,
0x1.97d829fde4e2ap-1, -0x1.1000p-47,
0x1.98f33e47a23a3p-1, 0x1.d000p-45,
0x1.9a0f170ca0604p-1, -0x1.8a40p-44,
0x1.9b2bb4d53ff89p-1, 0x1.55c0p-44,
0x1.9c49182a3f15bp-1, 0x1.6b80p-45,
0x1.9d674194bb8c5p-1, -0x1.c000p-49,
0x1.9e86319e3238ep-1, 0x1.7d00p-46,
0x1.9fa5e8d07f302p-1, 0x1.6400p-46,
0x1.a0c667b5de54dp-1, -0x1.5000p-48,
0x1.a1e7aed8eb8f6p-1, 0x1.9e00p-47,
0x1.a309bec4a2e27p-1, 0x1.ad80p-45,
0x1.a42c980460a5dp-1, -0x1.af00p-46,
0x1.a5503b23e259bp-1, 0x1.b600p-47,
0x1.a674a8af46213p-1, 0x1.8880p-44,
0x1.a799e1330b3a7p-1, 0x1.1200p-46,
0x1.a8bfe53c12e8dp-1, 0x1.6c00p-47,
0x1.a9e6b5579fcd2p-1, -0x1.9b80p-45,
0x1.ab0e521356fb8p-1, 0x1.b700p-45,
0x1.ac36bbfd3f381p-1, 0x1.9000p-50,
0x1.ad5ff3a3c2780p-1, 0x1.4000p-49,
0x1.ae89f995ad2a3p-1, -0x1.c900p-45,
0x1.afb4ce622f367p-1, 0x1.6500p-46,
0x1.b0e07298db790p-1, 0x1.fd40p-45,
0x1.b20ce6c9a89a9p-1, 0x1.2700p-46,
0x1.b33a2b84f1a4bp-1, 0x1.d470p-43,
0x1.b468415b747e7p-1, -0x1.8380p-44,
0x1.b59728de5593ap-1, 0x1.8000p-54,
0x1.b6c6e29f1c56ap-1, 0x1.ad00p-47,
0x1.b7f76f2fb5e50p-1, 0x1.e800p-50,
0x1.b928cf22749b2p-1, -0x1.4c00p-47,
0x1.ba5b030a10603p-1, -0x1.d700p-47,
0x1.bb8e0b79a6f66p-1, 0x1.d900p-47,
0x1.bcc1e904bc1ffp-1, 0x1.2a00p-47,
0x1.bdf69c3f3a16fp-1, -0x1.f780p-46,
0x1.bf2c25bd71db8p-1, -0x1.0a00p-46,
0x1.c06286141b2e9p-1, -0x1.1400p-46,
0x1.c199bdd8552e0p-1, 0x1.be00p-47,
0x1.c2d1cd9fa64eep-1, -0x1.9400p-47,
0x1.c40ab5fffd02fp-1, -0x1.ed00p-47,
0x1.c544778fafd15p-1, 0x1.9660p-44,
0x1.c67f12e57d0cbp-1, -0x1.a100p-46,
0x1.c7ba88988c1b6p-1, -0x1.8458p-42,
0x1.c8f6d9406e733p-1, -0x1.a480p-46,
0x1.ca3405751c4dfp-1, 0x1.b000p-51,
0x1.cb720dcef9094p-1, 0x1.1400p-47,
0x1.ccb0f2e6d1689p-1, 0x1.0200p-48,
0x1.cdf0b555dc412p-1, 0x1.3600p-48,
0x1.cf3155b5bab3bp-1, -0x1.6900p-47,
0x1.d072d4a0789bcp-1, 0x1.9a00p-47,
0x1.d1b532b08c8fap-1, -0x1.5e00p-46,
0x1.d2f87080d8a85p-1, 0x1.d280p-46,
0x1.d43c8eacaa203p-1, 0x1.1a00p-47,
0x1.d5818dcfba491p-1, 0x1.f000p-50,
0x1.d6c76e862e6a1p-1, -0x1.3a00p-47,
0x1.d80e316c9834ep-1, -0x1.cd80p-47,
0x1.d955d71ff6090p-1, 0x1.4c00p-48,
0x1.da9e603db32aep-1, 0x1.f900p-48,
0x1.dbe7cd63a8325p-1, 0x1.9800p-49,
0x1.dd321f301b445p-1, -0x1.5200p-48,
0x1.de7d5641c05bfp-1, -0x1.d700p-46,
0x1.dfc97337b9aecp-1, -0x1.6140p-46,
0x1.e11676b197d5ep-1, 0x1.b480p-47,
0x1.e264614f5a3e7p-1, 0x1.0ce0p-43,
0x1.e3b333b16ee5cp-1, 0x1.c680p-47,
0x1.e502ee78b3fb4p-1, -0x1.9300p-47,
0x1.e653924676d68p-1, -0x1.5000p-49,
0x1.e7a51fbc74c44p-1, -0x1.7f80p-47,
0x1.e8f7977cdb726p-1, -0x1.3700p-48,
0x1.ea4afa2a490e8p-1, 0x1.5d00p-49,
0x1.eb9f4867ccae4p-1, 0x1.61a0p-46,
0x1.ecf482d8e680dp-1, 0x1.5500p-48,
0x1.ee4aaa2188514p-1, 0x1.6400p-51,
0x1.efa1bee615a13p-1, -0x1.e800p-49,
0x1.f0f9c1cb64106p-1, -0x1.a880p-48,
0x1.f252b376bb963p-1, -0x1.c900p-45,
0x1.f3ac948dd7275p-1, 0x1.a000p-53,
0x1.f50765b6e4524p-1, -0x1.4f00p-48,
0x1.f6632798844fdp-1, 0x1.a800p-51,
0x1.f7bfdad9cbe38p-1, 0x1.abc0p-48,
0x1.f91d802243c82p-1, -0x1.4600p-50,
0x1.fa7c1819e908ep-1, -0x1.b0c0p-47,
0x1.fbdba3692d511p-1, -0x1.0e00p-51,
0x1.fd3c22b8f7194p-1, -0x1.0de8p-46,
0x1.fe9d96b2a23eep-1, 0x1.e430p-49,
0x1.0000000000000p+0, 0x0.0000p+0,
0x1.00b1afa5abcbep+0, -0x1.3400p-52,
0x1.0163da9fb3303p+0, -0x1.2170p-46,
0x1.02168143b0282p+0, 0x1.a400p-52,
0x1.02c9a3e77806cp+0, 0x1.f980p-49,
0x1.037d42e11bbcap+0, -0x1.7400p-51,
0x1.04315e86e7f89p+0, 0x1.8300p-50,
0x1.04e5f72f65467p+0, -0x1.a3f0p-46,
0x1.059b0d315855ap+0, -0x1.2840p-47,
0x1.0650a0e3c1f95p+0, 0x1.1600p-48,
0x1.0706b29ddf71ap+0, 0x1.5240p-46,
0x1.07bd42b72a82dp+0, -0x1.9a00p-49,
0x1.0874518759bd0p+0, 0x1.6400p-49,
0x1.092bdf66607c8p+0, -0x1.0780p-47,
0x1.09e3ecac6f383p+0, -0x1.8000p-54,
0x1.0a9c79b1f3930p+0, 0x1.fa00p-48,
0x1.0b5586cf988fcp+0, -0x1.ac80p-48,
0x1.0c0f145e46c8ap+0, 0x1.9c00p-50,
0x1.0cc922b724816p+0, 0x1.5200p-47,
0x1.0d83b23395dd8p+0, -0x1.ad00p-48,
0x1.0e3ec32d3d1f3p+0, 0x1.bac0p-46,
0x1.0efa55fdfa9a6p+0, -0x1.4e80p-47,
0x1.0fb66affed2f0p+0, -0x1.d300p-47,
0x1.1073028d7234bp+0, 0x1.1500p-48,
0x1.11301d0125b5bp+0, 0x1.c000p-49,
0x1.11edbab5e2af9p+0, 0x1.6bc0p-46,
0x1.12abdc06c31d5p+0, 0x1.8400p-49,
0x1.136a814f2047dp+0, -0x1.ed00p-47,
0x1.1429aaea92de9p+0, 0x1.8e00p-49,
0x1.14e95934f3138p+0, 0x1.b400p-49,
0x1.15a98c8a58e71p+0, 0x1.5300p-47,
0x1.166a45471c3dfp+0, 0x1.3380p-47,
0x1.172b83c7d5211p+0, 0x1.8d40p-45,
0x1.17ed48695bb9fp+0, -0x1.5d00p-47,
0x1.18af9388c8d93p+0, -0x1.c880p-46,
0x1.1972658375d66p+0, 0x1.1f00p-46,
0x1.1a35beb6fcba7p+0, 0x1.0480p-46,
0x1.1af99f81387e3p+0, -0x1.7390p-43,
0x1.1bbe084045d54p+0, 0x1.4e40p-45,
0x1.1c82f95281c43p+0, -0x1.a200p-47,
0x1.1d4873168b9b2p+0, 0x1.3800p-49,
0x1.1e0e75eb44031p+0, 0x1.ac00p-49,
0x1.1ed5022fcd938p+0, 0x1.1900p-47,
0x1.1f9c18438cdf7p+0, -0x1.b780p-46,
0x1.2063b88628d8fp+0, 0x1.d940p-45,
0x1.212be3578a81ep+0, 0x1.8000p-50,
0x1.21f49917ddd41p+0, 0x1.b340p-45,
0x1.22bdda2791323p+0, 0x1.9f80p-46,
0x1.2387a6e7561e7p+0, -0x1.9c80p-46,
0x1.2451ffb821427p+0, 0x1.2300p-47,
0x1.251ce4fb2a602p+0, -0x1.3480p-46,
0x1.25e85711eceb0p+0, 0x1.2700p-46,
0x1.26b4565e27d16p+0, 0x1.1d00p-46,
0x1.2780e341de00fp+0, 0x1.1ee0p-44,
0x1.284dfe1f5633ep+0, -0x1.4c00p-46,
0x1.291ba7591bb30p+0, -0x1.3d80p-46,
0x1.29e9df51fdf09p+0, 0x1.8b00p-47,
0x1.2ab8a66d10e9bp+0, -0x1.27c0p-45,
0x1.2b87fd0dada3ap+0, 0x1.a340p-45,
0x1.2c57e39771af9p+0, -0x1.0800p-46,
0x1.2d285a6e402d9p+0, -0x1.ed00p-47,
0x1.2df961f641579p+0, -0x1.4200p-48,
0x1.2ecafa93e2ecfp+0, -0x1.4980p-45,
0x1.2f9d24abd8822p+0, -0x1.6300p-46,
0x1.306fe0a31b625p+0, -0x1.2360p-44,
0x1.31432edeea50bp+0, -0x1.0df8p-40,
0x1.32170fc4cd7b8p+0, -0x1.2480p-45,
0x1.32eb83ba8e9a2p+0, -0x1.5980p-45,
0x1.33c08b2641766p+0, 0x1.ed00p-46,
0x1.3496266e3fa27p+0, -0x1.c000p-50,
0x1.356c55f929f0fp+0, -0x1.0d80p-44,
0x1.36431a2de88b9p+0, 0x1.2c80p-45,
0x1.371a7373aaa39p+0, 0x1.0600p-45,
0x1.37f26231e74fep+0, -0x1.6600p-46,
0x1.38cae6d05d838p+0, -0x1.ae00p-47,
0x1.39a401b713ec3p+0, -0x1.4720p-43,
0x1.3a7db34e5a020p+0, 0x1.8200p-47,
0x1.3b57fbfec6e95p+0, 0x1.e800p-44,
0x1.3c32dc313a8f2p+0, 0x1.f800p-49,
0x1.3d0e544ede122p+0, -0x1.7a00p-46,
0x1.3dea64c1234bbp+0, 0x1.6300p-45,
0x1.3ec70df1c4eccp+0, -0x1.8a60p-43,
0x1.3fa4504ac7e8cp+0, -0x1.cdc0p-44,
0x1.40822c367a0bbp+0, 0x1.5b80p-45,
0x1.4160a21f72e95p+0, 0x1.ec00p-46,
0x1.423fb27094646p+0, -0x1.3600p-46,
0x1.431f5d950a920p+0, 0x1.3980p-45,
0x1.43ffa3f84b9ebp+0, 0x1.a000p-48,
0x1.44e0860618919p+0, -0x1.6c00p-48,
0x1.45c2042a7d201p+0, -0x1.bc00p-47,
0x1.46a41ed1d0016p+0, -0x1.2800p-46,
0x1.4786d668b3326p+0, 0x1.0e00p-44,
0x1.486a2b5c13c00p+0, -0x1.d400p-45,
0x1.494e1e192af04p+0, 0x1.c200p-47,
0x1.4a32af0d7d372p+0, -0x1.e500p-46,
0x1.4b17dea6db801p+0, 0x1.7800p-47,
0x1.4bfdad53629e1p+0, -0x1.3800p-46,
0x1.4ce41b817c132p+0, 0x1.0800p-47,
0x1.4dcb299fddddbp+0, 0x1.c700p-45,
0x1.4eb2d81d8ab96p+0, -0x1.ce00p-46,
0x1.4f9b2769d2d02p+0, 0x1.9200p-46,
0x1.508417f4531c1p+0, -0x1.8c00p-47,
0x1.516daa2cf662ap+0, -0x1.a000p-48,
0x1.5257de83f51eap+0, 0x1.a080p-43,
0x1.5342b569d4edap+0, -0x1.6d80p-45,
0x1.542e2f4f6ac1ap+0, -0x1.2440p-44,
0x1.551a4ca5d94dbp+0, 0x1.83c0p-43,
0x1.56070dde9116bp+0, 0x1.4b00p-45,
0x1.56f4736b529dep+0, 0x1.15a0p-43,
0x1.57e27dbe2c40ep+0, -0x1.9e00p-45,
0x1.58d12d497c76fp+0, -0x1.3080p-45,
0x1.59c0827ff0b4cp+0, 0x1.dec0p-43,
0x1.5ab07dd485427p+0, -0x1.4000p-51,
0x1.5ba11fba87af4p+0, 0x1.0080p-44,
0x1.5c9268a59460bp+0, -0x1.6c80p-45,
0x1.5d84590998e3fp+0, 0x1.69a0p-43,
0x1.5e76f15ad20e1p+0, -0x1.b400p-46,
0x1.5f6a320dcebcap+0, 0x1.7700p-46,
0x1.605e1b976dcb8p+0, 0x1.6f80p-45,
0x1.6152ae6cdf715p+0, 0x1.1000p-47,
0x1.6247eb03a5531p+0, -0x1.5d00p-46,
0x1.633dd1d1929b5p+0, -0x1.2d00p-46,
0x1.6434634ccc313p+0, -0x1.a800p-49,
0x1.652b9febc8efap+0, -0x1.8600p-45,
0x1.6623882553397p+0, 0x1.1fe0p-40,
0x1.671c1c708328ep+0, -0x1.7200p-44,
0x1.68155d44ca97ep+0, 0x1.6800p-49,
0x1.690f4b19e9471p+0, -0x1.9780p-45,
};
/*
* exp2(x): compute the base 2 exponential of x
*
* Accuracy: Peak error < 0.503 ulp for normalized results.
*
* Method: (accurate tables)
*
* Reduce x:
* x = 2**k + y, for integer k and |y| <= 1/2.
* Thus we have exp2(x) = 2**k * exp2(y).
*
* Reduce y:
* y = i/TBLSIZE + z - eps[i] for integer i near y * TBLSIZE.
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z - eps[i]),
* with |z - eps[i]| <= 2**-9 + 2**-39 for the table used.
*
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z - eps[i]) via
* a degree-5 minimax polynomial with maximum error under 1.3 * 2**-61.
* The values in exp2t[] and eps[] are chosen such that
* exp2t[i] = exp2(i/TBLSIZE + eps[i]), and eps[i] is a small offset such
* that exp2t[i] is accurate to 2**-64.
*
* Note that the range of i is +-TBLSIZE/2, so we actually index the tables
* by i0 = i + TBLSIZE/2. For cache efficiency, exp2t[] and eps[] are
* virtual tables, interleaved in the real table tbl[].
*
* This method is due to Gal, with many details due to Gal and Bachelis:
*
* Gal, S. and Bachelis, B. An Accurate Elementary Mathematical Library
* for the IEEE Floating Point Standard. TOMS 17(1), 26-46 (1991).
*/
double
exp2(double x)
{
double r, t, twopk, twopkp1000, z;
uint32_t hx, ix, lx, i0;
int k;
/* Filter out exceptional cases. */
GET_HIGH_WORD(hx,x);
ix = hx & 0x7fffffff; /* high word of |x| */
if(ix >= 0x40900000) { /* |x| >= 1024 */
if(ix >= 0x7ff00000) {
GET_LOW_WORD(lx,x);
if(((ix & 0xfffff) | lx) != 0 || (hx & 0x80000000) == 0)
return (x + x); /* x is NaN or +Inf */
else
return (0.0); /* x is -Inf */
}
if(x >= 0x1.0p10)
return (huge * huge); /* overflow */
if(x <= -0x1.0ccp10)
return (twom1000 * twom1000); /* underflow */
} else if (ix < 0x3c900000) { /* |x| < 0x1p-54 */
return (1.0 + x);
}
/* Reduce x, computing z, i0, and k. */
STRICT_ASSIGN(double, t, x + redux);
GET_LOW_WORD(i0, t);
i0 += TBLSIZE / 2;
k = (i0 >> TBLBITS) << 20;
i0 = (i0 & (TBLSIZE - 1)) << 1;
t -= redux;
z = x - t;
/* Compute r = exp2(y) = exp2t[i0] * p(z - eps[i]). */
t = tbl[i0]; /* exp2t[i0] */
z -= tbl[i0 + 1]; /* eps[i0] */
if (k >= -1021 << 20)
INSERT_WORDS(twopk, 0x3ff00000 + k, 0);
else
INSERT_WORDS(twopkp1000, 0x3ff00000 + k + (1000 << 20), 0);
r = t + t * z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * P5))));
/* Scale by 2**(k>>20). */
if(k >= -1021 << 20) {
if (k == 1024 << 20)
return (r * 2.0 * 0x1p1023);
return (r * twopk);
} else {
return (r * twopkp1000 * twom1000);
}
}
#ifdef notyet
#if (LDBL_MANT_DIG == 53)
__weak_reference(exp2, exp2l);
#endif
#endif
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