Jmat.Real.dawson

Average Error: 29.7 → 0.0

Time: 35.7s

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\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -753.9125604904361352964770048856735229492 \lor \neg \left(x \le 693.3656209634430069854715839028358459473\right):\\ \;\;\;\;\mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, 0.5 \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \frac{x}{\mathsf{fma}\left(-2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{12}, \left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) - 0.01400054419999999938406531896362139377743 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) - 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{6}\right)\right)}\\ \end{array}\]

C

double f(double x) {
        double r221041 = 1.0;
        double r221042 = 0.1049934947;
        double r221043 = x;
        double r221044 = r221043 * r221043;
        double r221045 = r221042 * r221044;
        double r221046 = r221041 + r221045;
        double r221047 = 0.0424060604;
        double r221048 = r221044 * r221044;
        double r221049 = r221047 * r221048;
        double r221050 = r221046 + r221049;
        double r221051 = 0.0072644182;
        double r221052 = r221048 * r221044;
        double r221053 = r221051 * r221052;
        double r221054 = r221050 + r221053;
        double r221055 = 0.0005064034;
        double r221056 = r221052 * r221044;
        double r221057 = r221055 * r221056;
        double r221058 = r221054 + r221057;
        double r221059 = 0.0001789971;
        double r221060 = r221056 * r221044;
        double r221061 = r221059 * r221060;
        double r221062 = r221058 + r221061;
        double r221063 = 0.7715471019;
        double r221064 = r221063 * r221044;
        double r221065 = r221041 + r221064;
        double r221066 = 0.2909738639;
        double r221067 = r221066 * r221048;
        double r221068 = r221065 + r221067;
        double r221069 = 0.0694555761;
        double r221070 = r221069 * r221052;
        double r221071 = r221068 + r221070;
        double r221072 = 0.0140005442;
        double r221073 = r221072 * r221056;
        double r221074 = r221071 + r221073;
        double r221075 = 0.0008327945;
        double r221076 = r221075 * r221060;
        double r221077 = r221074 + r221076;
        double r221078 = 2.0;
        double r221079 = r221078 * r221059;
        double r221080 = r221060 * r221044;
        double r221081 = r221079 * r221080;
        double r221082 = r221077 + r221081;
        double r221083 = r221062 / r221082;
        double r221084 = r221083 * r221043;
        return r221084;
}

↓

double f(double x) {
        double r221085 = x;
        double r221086 = -753.9125604904361;
        bool r221087 = r221085 <= r221086;
        double r221088 = 693.365620963443;
        bool r221089 = r221085 <= r221088;
        double r221090 = !r221089;
        bool r221091 = r221087 || r221090;
        double r221092 = 0.2514179000665375;
        double r221093 = 1.0;
        double r221094 = 3.0;
        double r221095 = pow(r221085, r221094);
        double r221096 = r221093 / r221095;
        double r221097 = 0.15298196345929327;
        double r221098 = 5.0;
        double r221099 = pow(r221085, r221098);
        double r221100 = r221093 / r221099;
        double r221101 = 0.5;
        double r221102 = r221093 / r221085;
        double r221103 = r221101 * r221102;
        double r221104 = fma(r221097, r221100, r221103);
        double r221105 = fma(r221092, r221096, r221104);
        double r221106 = 0.0424060604;
        double r221107 = 4.0;
        double r221108 = pow(r221085, r221107);
        double r221109 = 0.1049934947;
        double r221110 = r221109 * r221085;
        double r221111 = 1.0;
        double r221112 = fma(r221110, r221085, r221111);
        double r221113 = log1p(r221112);
        double r221114 = expm1(r221113);
        double r221115 = fma(r221106, r221108, r221114);
        double r221116 = -r221115;
        double r221117 = 0.0072644182;
        double r221118 = 6.0;
        double r221119 = pow(r221085, r221118);
        double r221120 = 2.0;
        double r221121 = pow(r221085, r221120);
        double r221122 = r221085 * r221095;
        double r221123 = r221121 * r221122;
        double r221124 = r221121 * r221123;
        double r221125 = 0.0005064034;
        double r221126 = r221124 * r221125;
        double r221127 = fma(r221117, r221119, r221126);
        double r221128 = -r221127;
        double r221129 = r221116 + r221128;
        double r221130 = 0.0001789971;
        double r221131 = r221130 * r221119;
        double r221132 = r221131 * r221108;
        double r221133 = r221129 - r221132;
        double r221134 = 2.0;
        double r221135 = r221134 * r221130;
        double r221136 = -r221135;
        double r221137 = 12.0;
        double r221138 = pow(r221085, r221137);
        double r221139 = 0.0694555761;
        double r221140 = 0.2909738639;
        double r221141 = 0.7715471019;
        double r221142 = r221141 * r221085;
        double r221143 = fma(r221142, r221085, r221111);
        double r221144 = fma(r221140, r221108, r221143);
        double r221145 = fma(r221139, r221119, r221144);
        double r221146 = -r221145;
        double r221147 = 0.0140005442;
        double r221148 = r221147 * r221124;
        double r221149 = r221146 - r221148;
        double r221150 = 0.0008327945;
        double r221151 = r221121 * r221121;
        double r221152 = r221151 * r221119;
        double r221153 = r221150 * r221152;
        double r221154 = r221149 - r221153;
        double r221155 = fma(r221136, r221138, r221154);
        double r221156 = r221085 / r221155;
        double r221157 = r221133 * r221156;
        double r221158 = r221091 ? r221105 : r221157;
        return r221158;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -753.9125604904361 or 693.365620963443 < x

   1. Initial program 59.1

      \[\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 frac-2neg 59.1

      \[\leadsto \color{blue}{\frac{-\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)}{-\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. Simplified 59.1

      \[\leadsto \frac{\color{blue}{\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}}}{-\left(\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot x\]

   5. Simplified 59.0

      \[\leadsto \frac{\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}}{\color{blue}{\left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right), \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right)\right) - \left(\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot {x}^{8}\right) \cdot {x}^{4}}} \cdot x\]
   6. Using strategy rm

   7. Applied div-inv 59.0

      \[\leadsto \color{blue}{\left(\left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \frac{1}{\left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right), \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right)\right) - \left(\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot {x}^{8}\right) \cdot {x}^{4}}\right)} \cdot x\]

   8. Applied associate-*l* 59.0

      \[\leadsto \color{blue}{\left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \left(\frac{1}{\left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right), \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right)\right) - \left(\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot {x}^{8}\right) \cdot {x}^{4}} \cdot x\right)}\]

   9. Simplified 59.1

      \[\leadsto \left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \color{blue}{\frac{x}{\mathsf{fma}\left(-2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{12}, \left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) - 0.01400054419999999938406531896362139377743 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) - 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{6}\right)\right)}}\]

   10. Taylor expanded around inf 0.0

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

   11. Simplified 0.0

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

   if -753.9125604904361 < x < 693.365620963443

   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 frac-2neg 0.0

      \[\leadsto \color{blue}{\frac{-\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)}{-\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. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}}}{-\left(\left(\left(\left(\left(\left(1 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot x\]

   5. Simplified 0.0

      \[\leadsto \frac{\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}}{\color{blue}{\left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right), \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right)\right) - \left(\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot {x}^{8}\right) \cdot {x}^{4}}} \cdot x\]
   6. Using strategy rm

   7. Applied div-inv 0.0

      \[\leadsto \color{blue}{\left(\left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \frac{1}{\left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right), \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right)\right) - \left(\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot {x}^{8}\right) \cdot {x}^{4}}\right)} \cdot x\]

   8. Applied associate-*l* 0.0

      \[\leadsto \color{blue}{\left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \left(\frac{1}{\left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right), \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right)\right) - \left(\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) \cdot {x}^{8}\right) \cdot {x}^{4}} \cdot x\right)}\]

   9. Simplified 0.0

      \[\leadsto \left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \color{blue}{\frac{x}{\mathsf{fma}\left(-2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{12}, \left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) - 0.01400054419999999938406531896362139377743 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) - 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{6}\right)\right)}}\]
   10. Using strategy rm

   11. Applied expm1-log1p-u 0.0

       \[\leadsto \left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \frac{x}{\mathsf{fma}\left(-2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{12}, \left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) - 0.01400054419999999938406531896362139377743 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) - 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{6}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -753.9125604904361352964770048856735229492 \lor \neg \left(x \le 693.3656209634430069854715839028358459473\right):\\ \;\;\;\;\mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, 0.5 \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right) \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right)\right)\right) - \left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6}\right) \cdot {x}^{4}\right) \cdot \frac{x}{\mathsf{fma}\left(-2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{12}, \left(\left(-\mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right) - 0.01400054419999999938406531896362139377743 \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) - 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{6}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))