Jmat.Real.dawson

Average Error: 29.2 → 0.0

Time: 1.6m

Precision: 64

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -70734291355216568647680 \lor \neg \left(x \le \frac{4097441582874279}{4398046511104}\right):\\ \;\;\;\;1 \cdot \left(\frac{1132285561053931}{4503599627370496} \cdot \frac{1}{{x}^{3}} + \left(\frac{43060594601855}{281474976710656} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x \cdot \left(\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot \frac{825478423409049}{4611686018427387904} + \left(\left(\frac{8375303961237363 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{1152921504606846976} + 1\right) + {x}^{2} \cdot \left(\frac{3782789308857969}{36028797018963968} + \frac{763919671262763}{18014398509481984} \cdot {x}^{2}\right)\right)\right) + \frac{583843369866023 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)}{1152921504606846976}\right)}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right)\right) \cdot \left(2 \cdot \frac{825478423409049}{4611686018427387904}\right) + \left(\left(\frac{2502400853142105 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{36028797018963968} + 1\right) + {x}^{2} \cdot \left(\frac{1737369620307813}{2251799813685248} + \frac{2620859570069187}{9007199254740992} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{2017691060547333}{144115188075855872} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{7681173503746455}{9223372036854775808} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r129004 = 1.0;
        double r129005 = 0.1049934947;
        double r129006 = x;
        double r129007 = r129006 * r129006;
        double r129008 = r129005 * r129007;
        double r129009 = r129004 + r129008;
        double r129010 = 0.0424060604;
        double r129011 = r129007 * r129007;
        double r129012 = r129010 * r129011;
        double r129013 = r129009 + r129012;
        double r129014 = 0.0072644182;
        double r129015 = r129011 * r129007;
        double r129016 = r129014 * r129015;
        double r129017 = r129013 + r129016;
        double r129018 = 0.0005064034;
        double r129019 = r129015 * r129007;
        double r129020 = r129018 * r129019;
        double r129021 = r129017 + r129020;
        double r129022 = 0.0001789971;
        double r129023 = r129019 * r129007;
        double r129024 = r129022 * r129023;
        double r129025 = r129021 + r129024;
        double r129026 = 0.7715471019;
        double r129027 = r129026 * r129007;
        double r129028 = r129004 + r129027;
        double r129029 = 0.2909738639;
        double r129030 = r129029 * r129011;
        double r129031 = r129028 + r129030;
        double r129032 = 0.0694555761;
        double r129033 = r129032 * r129015;
        double r129034 = r129031 + r129033;
        double r129035 = 0.0140005442;
        double r129036 = r129035 * r129019;
        double r129037 = r129034 + r129036;
        double r129038 = 0.0008327945;
        double r129039 = r129038 * r129023;
        double r129040 = r129037 + r129039;
        double r129041 = 2.0;
        double r129042 = r129041 * r129022;
        double r129043 = r129023 * r129007;
        double r129044 = r129042 * r129043;
        double r129045 = r129040 + r129044;
        double r129046 = r129025 / r129045;
        double r129047 = r129046 * r129006;
        return r129047;
}

↓

double f(double x) {
        double r129048 = x;
        double r129049 = -7.073429135521657e+22;
        bool r129050 = r129048 <= r129049;
        double r129051 = 4097441582874279.0;
        double r129052 = 4398046511104.0;
        double r129053 = r129051 / r129052;
        bool r129054 = r129048 <= r129053;
        double r129055 = !r129054;
        bool r129056 = r129050 || r129055;
        double r129057 = 1.0;
        double r129058 = 1132285561053931.0;
        double r129059 = 4503599627370496.0;
        double r129060 = r129058 / r129059;
        double r129061 = 3.0;
        double r129062 = pow(r129048, r129061);
        double r129063 = r129057 / r129062;
        double r129064 = r129060 * r129063;
        double r129065 = 43060594601855.0;
        double r129066 = 281474976710656.0;
        double r129067 = r129065 / r129066;
        double r129068 = 5.0;
        double r129069 = pow(r129048, r129068);
        double r129070 = r129057 / r129069;
        double r129071 = r129067 * r129070;
        double r129072 = 1.0;
        double r129073 = 2.0;
        double r129074 = r129072 / r129073;
        double r129075 = r129057 / r129048;
        double r129076 = r129074 * r129075;
        double r129077 = r129071 + r129076;
        double r129078 = r129064 + r129077;
        double r129079 = r129057 * r129078;
        double r129080 = 2.0;
        double r129081 = pow(r129048, r129080);
        double r129082 = r129048 * r129062;
        double r129083 = r129081 * r129082;
        double r129084 = r129081 * r129083;
        double r129085 = r129081 * r129084;
        double r129086 = 825478423409049.0;
        double r129087 = 4.611686018427388e+18;
        double r129088 = r129086 / r129087;
        double r129089 = r129085 * r129088;
        double r129090 = 8375303961237363.0;
        double r129091 = r129090 * r129083;
        double r129092 = 1.152921504606847e+18;
        double r129093 = r129091 / r129092;
        double r129094 = r129093 + r129072;
        double r129095 = 3782789308857969.0;
        double r129096 = 3.602879701896397e+16;
        double r129097 = r129095 / r129096;
        double r129098 = 763919671262763.0;
        double r129099 = 18014398509481984.0;
        double r129100 = r129098 / r129099;
        double r129101 = r129100 * r129081;
        double r129102 = r129097 + r129101;
        double r129103 = r129081 * r129102;
        double r129104 = r129094 + r129103;
        double r129105 = r129089 + r129104;
        double r129106 = 583843369866023.0;
        double r129107 = r129106 * r129084;
        double r129108 = r129107 / r129092;
        double r129109 = r129105 + r129108;
        double r129110 = r129048 * r129109;
        double r129111 = r129081 * r129085;
        double r129112 = r129073 * r129088;
        double r129113 = r129111 * r129112;
        double r129114 = 2502400853142105.0;
        double r129115 = r129114 * r129083;
        double r129116 = r129115 / r129096;
        double r129117 = r129116 + r129072;
        double r129118 = 1737369620307813.0;
        double r129119 = 2251799813685248.0;
        double r129120 = r129118 / r129119;
        double r129121 = 2620859570069187.0;
        double r129122 = 9007199254740992.0;
        double r129123 = r129121 / r129122;
        double r129124 = r129123 * r129081;
        double r129125 = r129120 + r129124;
        double r129126 = r129081 * r129125;
        double r129127 = r129117 + r129126;
        double r129128 = r129113 + r129127;
        double r129129 = 2017691060547333.0;
        double r129130 = 1.4411518807585587e+17;
        double r129131 = r129129 / r129130;
        double r129132 = r129048 * r129048;
        double r129133 = r129132 * r129048;
        double r129134 = r129133 * r129133;
        double r129135 = r129131 * r129134;
        double r129136 = 7681173503746455.0;
        double r129137 = 9.223372036854776e+18;
        double r129138 = r129136 / r129137;
        double r129139 = r129132 * r129132;
        double r129140 = r129139 * r129132;
        double r129141 = r129140 * r129132;
        double r129142 = r129138 * r129141;
        double r129143 = r129135 + r129142;
        double r129144 = r129081 * r129143;
        double r129145 = r129128 + r129144;
        double r129146 = r129110 / r129145;
        double r129147 = r129057 * r129146;
        double r129148 = r129056 ? r129079 : r129147;
        return r129148;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -7.073429135521657e+22 or 931.6503526120594 < x

   1. Initial program 61.3

      \[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 61.3

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}} \cdot x\]

   4. Applied *-un-lft-identity 61.3

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}}{1 \cdot \left(\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot x\]

   5. Applied times-frac 61.3

      \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}\right)} \cdot x\]

   6. Applied associate-*l* 61.3

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\right)}\]

   7. Simplified 61.3

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{x \cdot \left(\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot \frac{825478423409049}{4611686018427387904} + \left(\left(\frac{8375303961237363 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{1152921504606846976} + 1\right) + {x}^{2} \cdot \left(\frac{3782789308857969}{36028797018963968} + \frac{763919671262763}{18014398509481984} \cdot {x}^{2}\right)\right)\right) + \frac{583843369866023 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)}{1152921504606846976}\right)}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right)\right) \cdot \left(2 \cdot \frac{825478423409049}{4611686018427387904}\right) + \left(\left(\frac{2502400853142105 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{36028797018963968} + 1\right) + {x}^{2} \cdot \left(\frac{1737369620307813}{2251799813685248} + \frac{2620859570069187}{9007199254740992} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{2017691060547333}{144115188075855872} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{7681173503746455}{9223372036854775808} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}}\]

   8. Taylor expanded around inf 0.0

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\left(0.2514179000665375252054900556686334311962 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592932686700805788859724998474 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\right)}\]

   9. Simplified 0.0

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\left(\frac{1132285561053931}{4503599627370496} \cdot \frac{1}{{x}^{3}} + \left(\frac{43060594601855}{281474976710656} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

   if -7.073429135521657e+22 < x < 931.6503526120594

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 0.0

      \[\leadsto \frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}} \cdot x\]

   4. Applied *-un-lft-identity 0.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}}{1 \cdot \left(\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot x\]

   5. Applied times-frac 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}\right)} \cdot x\]

   6. Applied associate-*l* 0.0

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\right)}\]

   7. Simplified 0.0

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{x \cdot \left(\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot \frac{825478423409049}{4611686018427387904} + \left(\left(\frac{8375303961237363 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{1152921504606846976} + 1\right) + {x}^{2} \cdot \left(\frac{3782789308857969}{36028797018963968} + \frac{763919671262763}{18014398509481984} \cdot {x}^{2}\right)\right)\right) + \frac{583843369866023 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)}{1152921504606846976}\right)}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right)\right) \cdot \left(2 \cdot \frac{825478423409049}{4611686018427387904}\right) + \left(\left(\frac{2502400853142105 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{36028797018963968} + 1\right) + {x}^{2} \cdot \left(\frac{1737369620307813}{2251799813685248} + \frac{2620859570069187}{9007199254740992} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{2017691060547333}{144115188075855872} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{7681173503746455}{9223372036854775808} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -70734291355216568647680 \lor \neg \left(x \le \frac{4097441582874279}{4398046511104}\right):\\ \;\;\;\;1 \cdot \left(\frac{1132285561053931}{4503599627370496} \cdot \frac{1}{{x}^{3}} + \left(\frac{43060594601855}{281474976710656} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x \cdot \left(\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot \frac{825478423409049}{4611686018427387904} + \left(\left(\frac{8375303961237363 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{1152921504606846976} + 1\right) + {x}^{2} \cdot \left(\frac{3782789308857969}{36028797018963968} + \frac{763919671262763}{18014398509481984} \cdot {x}^{2}\right)\right)\right) + \frac{583843369866023 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)}{1152921504606846976}\right)}{\left(\left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right)\right) \cdot \left(2 \cdot \frac{825478423409049}{4611686018427387904}\right) + \left(\left(\frac{2502400853142105 \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)}{36028797018963968} + 1\right) + {x}^{2} \cdot \left(\frac{1737369620307813}{2251799813685248} + \frac{2620859570069187}{9007199254740992} \cdot {x}^{2}\right)\right)\right) + {x}^{2} \cdot \left(\frac{2017691060547333}{144115188075855872} \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \frac{7681173503746455}{9223372036854775808} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.042406060400000001 (* (* x x) (* x x)))) (* 0.00726441819999999999 (* (* (* x x) (* x x)) (* x x)))) (* 5.0640340000000002e-4 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 1.789971e-4 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.77154710189999998 (* x x))) (* 0.29097386390000002 (* (* x x) (* x x)))) (* 0.069455576099999999 (* (* (* x x) (* x x)) (* x x)))) (* 0.014000544199999999 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 8.32794500000000044e-4 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 1.789971e-4) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))