Average Error: 29.6 → 0.0
Time: 30.4s
Precision: 64
\[\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 -656.5908303529681688814889639616012573242 \lor \neg \left(x \le 735244.049645020510070025920867919921875\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}:\\ \;\;\;\;\frac{1}{\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}} \cdot \frac{x}{\frac{-\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}{\left(-{x}^{4}\right) \cdot \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right) + \left(-\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\ \end{array}\]
\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 -656.5908303529681688814889639616012573242 \lor \neg \left(x \le 735244.049645020510070025920867919921875\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}:\\
\;\;\;\;\frac{1}{\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}} \cdot \frac{x}{\frac{-\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}{\left(-{x}^{4}\right) \cdot \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right) + \left(-\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\

\end{array}
double f(double x) {
        double r256311 = 1.0;
        double r256312 = 0.1049934947;
        double r256313 = x;
        double r256314 = r256313 * r256313;
        double r256315 = r256312 * r256314;
        double r256316 = r256311 + r256315;
        double r256317 = 0.0424060604;
        double r256318 = r256314 * r256314;
        double r256319 = r256317 * r256318;
        double r256320 = r256316 + r256319;
        double r256321 = 0.0072644182;
        double r256322 = r256318 * r256314;
        double r256323 = r256321 * r256322;
        double r256324 = r256320 + r256323;
        double r256325 = 0.0005064034;
        double r256326 = r256322 * r256314;
        double r256327 = r256325 * r256326;
        double r256328 = r256324 + r256327;
        double r256329 = 0.0001789971;
        double r256330 = r256326 * r256314;
        double r256331 = r256329 * r256330;
        double r256332 = r256328 + r256331;
        double r256333 = 0.7715471019;
        double r256334 = r256333 * r256314;
        double r256335 = r256311 + r256334;
        double r256336 = 0.2909738639;
        double r256337 = r256336 * r256318;
        double r256338 = r256335 + r256337;
        double r256339 = 0.0694555761;
        double r256340 = r256339 * r256322;
        double r256341 = r256338 + r256340;
        double r256342 = 0.0140005442;
        double r256343 = r256342 * r256326;
        double r256344 = r256341 + r256343;
        double r256345 = 0.0008327945;
        double r256346 = r256345 * r256330;
        double r256347 = r256344 + r256346;
        double r256348 = 2.0;
        double r256349 = r256348 * r256329;
        double r256350 = r256330 * r256314;
        double r256351 = r256349 * r256350;
        double r256352 = r256347 + r256351;
        double r256353 = r256332 / r256352;
        double r256354 = r256353 * r256313;
        return r256354;
}
↓
double f(double x) {
        double r256355 = x;
        double r256356 = -656.5908303529682;
        bool r256357 = r256355 <= r256356;
        double r256358 = 735244.0496450205;
        bool r256359 = r256355 <= r256358;
        double r256360 = !r256359;
        bool r256361 = r256357 || r256360;
        double r256362 = 0.2514179000665375;
        double r256363 = 1.0;
        double r256364 = 3.0;
        double r256365 = pow(r256355, r256364);
        double r256366 = r256363 / r256365;
        double r256367 = 0.15298196345929327;
        double r256368 = 5.0;
        double r256369 = pow(r256355, r256368);
        double r256370 = r256363 / r256369;
        double r256371 = 0.5;
        double r256372 = r256363 / r256355;
        double r256373 = r256371 * r256372;
        double r256374 = fma(r256367, r256370, r256373);
        double r256375 = fma(r256362, r256366, r256374);
        double r256376 = r256355 * r256355;
        double r256377 = pow(r256376, r256364);
        double r256378 = r256377 * r256365;
        double r256379 = r256355 * r256378;
        double r256380 = 0.0008327945;
        double r256381 = 2.0;
        double r256382 = 0.0001789971;
        double r256383 = r256381 * r256382;
        double r256384 = r256376 * r256383;
        double r256385 = r256380 + r256384;
        double r256386 = r256379 * r256385;
        double r256387 = 0.2909738639;
        double r256388 = r256387 * r256355;
        double r256389 = 0.7715471019;
        double r256390 = r256389 * r256355;
        double r256391 = 1.0;
        double r256392 = fma(r256390, r256355, r256391);
        double r256393 = fma(r256388, r256365, r256392);
        double r256394 = r256386 + r256393;
        double r256395 = 6.0;
        double r256396 = pow(r256355, r256395);
        double r256397 = 0.0694555761;
        double r256398 = 0.0140005442;
        double r256399 = r256376 * r256398;
        double r256400 = r256397 + r256399;
        double r256401 = r256396 * r256400;
        double r256402 = r256394 + r256401;
        double r256403 = sqrt(r256402);
        double r256404 = r256363 / r256403;
        double r256405 = -r256403;
        double r256406 = 4.0;
        double r256407 = pow(r256355, r256406);
        double r256408 = -r256407;
        double r256409 = 0.0072644182;
        double r256410 = r256355 * r256409;
        double r256411 = 0.0424060604;
        double r256412 = fma(r256355, r256410, r256411);
        double r256413 = r256408 * r256412;
        double r256414 = pow(r256376, r256406);
        double r256415 = 0.0005064034;
        double r256416 = r256376 * r256382;
        double r256417 = r256415 + r256416;
        double r256418 = 0.1049934947;
        double r256419 = r256418 * r256355;
        double r256420 = fma(r256419, r256355, r256391);
        double r256421 = fma(r256414, r256417, r256420);
        double r256422 = -r256421;
        double r256423 = r256413 + r256422;
        double r256424 = r256405 / r256423;
        double r256425 = r256355 / r256424;
        double r256426 = r256404 * r256425;
        double r256427 = r256361 ? r256375 : r256426;
        return r256427;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -656.5908303529682 or 735244.0496450205 < x

    1. Initial program 59.9

      \[\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. Simplified59.8

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -656.5908303529682 < x < 735244.0496450205

    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. Simplified0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied times-frac0.0

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

    7. Applied *-un-lft-identity0.0

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

    8. Applied times-frac0.0

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

    9. Simplified0.0

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

    11. Applied frac-2neg0.0

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

    12. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -656.5908303529681688814889639616012573242 \lor \neg \left(x \le 735244.049645020510070025920867919921875\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}:\\ \;\;\;\;\frac{1}{\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}} \cdot \frac{x}{\frac{-\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}{\left(-{x}^{4}\right) \cdot \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right) + \left(-\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))