Average Error: 28.8 → 0.0
Time: 1.2m
Precision: 64
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -946613766.6812615:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 670.0218683925953:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(2, 0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \mathsf{fma}\left(0.0140005442, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0694555761, \mathsf{fma}\left(0.2909738639, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)} \cdot \left(\left(\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0005064034\right) + 0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]
\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \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) + 0.0001789971 \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 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \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) + 0.0008327945 \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 0.0001789971\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 -946613766.6812615:\\
\;\;\;\;\frac{0.5}{x} + \left(\frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x}\right)\\

\mathbf{elif}\;x \le 670.0218683925953:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(2, 0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \mathsf{fma}\left(0.0140005442, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0694555761, \mathsf{fma}\left(0.2909738639, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)} \cdot \left(\left(\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0005064034\right) + 0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5}{x} + \left(\frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x}\right)\\

\end{array}
double f(double x) {
        double r6402509 = 1.0;
        double r6402510 = 0.1049934947;
        double r6402511 = x;
        double r6402512 = r6402511 * r6402511;
        double r6402513 = r6402510 * r6402512;
        double r6402514 = r6402509 + r6402513;
        double r6402515 = 0.0424060604;
        double r6402516 = r6402512 * r6402512;
        double r6402517 = r6402515 * r6402516;
        double r6402518 = r6402514 + r6402517;
        double r6402519 = 0.0072644182;
        double r6402520 = r6402516 * r6402512;
        double r6402521 = r6402519 * r6402520;
        double r6402522 = r6402518 + r6402521;
        double r6402523 = 0.0005064034;
        double r6402524 = r6402520 * r6402512;
        double r6402525 = r6402523 * r6402524;
        double r6402526 = r6402522 + r6402525;
        double r6402527 = 0.0001789971;
        double r6402528 = r6402524 * r6402512;
        double r6402529 = r6402527 * r6402528;
        double r6402530 = r6402526 + r6402529;
        double r6402531 = 0.7715471019;
        double r6402532 = r6402531 * r6402512;
        double r6402533 = r6402509 + r6402532;
        double r6402534 = 0.2909738639;
        double r6402535 = r6402534 * r6402516;
        double r6402536 = r6402533 + r6402535;
        double r6402537 = 0.0694555761;
        double r6402538 = r6402537 * r6402520;
        double r6402539 = r6402536 + r6402538;
        double r6402540 = 0.0140005442;
        double r6402541 = r6402540 * r6402524;
        double r6402542 = r6402539 + r6402541;
        double r6402543 = 0.0008327945;
        double r6402544 = r6402543 * r6402528;
        double r6402545 = r6402542 + r6402544;
        double r6402546 = 2.0;
        double r6402547 = r6402546 * r6402527;
        double r6402548 = r6402528 * r6402512;
        double r6402549 = r6402547 * r6402548;
        double r6402550 = r6402545 + r6402549;
        double r6402551 = r6402530 / r6402550;
        double r6402552 = r6402551 * r6402511;
        return r6402552;
}
↓
double f(double x) {
        double r6402553 = x;
        double r6402554 = -946613766.6812615;
        bool r6402555 = r6402553 <= r6402554;
        double r6402556 = 0.5;
        double r6402557 = r6402556 / r6402553;
        double r6402558 = 0.15298196345929327;
        double r6402559 = r6402553 * r6402553;
        double r6402560 = r6402559 * r6402559;
        double r6402561 = r6402553 * r6402560;
        double r6402562 = r6402558 / r6402561;
        double r6402563 = 0.2514179000665375;
        double r6402564 = r6402559 * r6402553;
        double r6402565 = r6402563 / r6402564;
        double r6402566 = r6402562 + r6402565;
        double r6402567 = r6402557 + r6402566;
        double r6402568 = 670.0218683925953;
        bool r6402569 = r6402553 <= r6402568;
        double r6402570 = 2.0;
        double r6402571 = 0.0001789971;
        double r6402572 = r6402559 * r6402560;
        double r6402573 = r6402572 * r6402572;
        double r6402574 = r6402571 * r6402573;
        double r6402575 = 0.0140005442;
        double r6402576 = r6402560 * r6402560;
        double r6402577 = 0.0008327945;
        double r6402578 = r6402577 * r6402559;
        double r6402579 = r6402576 * r6402578;
        double r6402580 = fma(r6402575, r6402576, r6402579);
        double r6402581 = 0.0694555761;
        double r6402582 = 0.2909738639;
        double r6402583 = 0.7715471019;
        double r6402584 = 1.0;
        double r6402585 = fma(r6402559, r6402583, r6402584);
        double r6402586 = fma(r6402582, r6402560, r6402585);
        double r6402587 = fma(r6402572, r6402581, r6402586);
        double r6402588 = r6402580 + r6402587;
        double r6402589 = fma(r6402570, r6402574, r6402588);
        double r6402590 = r6402553 / r6402589;
        double r6402591 = 0.0072644182;
        double r6402592 = r6402591 * r6402572;
        double r6402593 = 0.1049934947;
        double r6402594 = r6402559 * r6402593;
        double r6402595 = r6402584 + r6402594;
        double r6402596 = 0.0424060604;
        double r6402597 = r6402596 * r6402560;
        double r6402598 = r6402595 + r6402597;
        double r6402599 = r6402592 + r6402598;
        double r6402600 = r6402559 * r6402572;
        double r6402601 = 0.0005064034;
        double r6402602 = r6402600 * r6402601;
        double r6402603 = r6402599 + r6402602;
        double r6402604 = r6402600 * r6402559;
        double r6402605 = r6402571 * r6402604;
        double r6402606 = r6402603 + r6402605;
        double r6402607 = r6402590 * r6402606;
        double r6402608 = r6402569 ? r6402607 : r6402567;
        double r6402609 = r6402555 ? r6402567 : r6402608;
        return r6402609;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -946613766.6812615 or 670.0218683925953 < x

    1. Initial program 58.8

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.15298196345929327 \cdot \frac{1}{{x}^{5}} + \left(0.2514179000665375 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)} + \frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}\right) + \frac{0.5}{x}}\]

    if -946613766.6812615 < x < 670.0218683925953

    1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
    2. Using strategy rm

    3. Applied div-inv0.0

      \[\leadsto \color{blue}{\left(\left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{1}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}\right)} \cdot x\]

    4. Applied associate-*l*0.0

      \[\leadsto \color{blue}{\left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\frac{1}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\right)}\]

    5. Simplified0.0

      \[\leadsto \left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\frac{x}{\mathsf{fma}\left(2, \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0001789971, \mathsf{fma}\left(0.0140005442, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(0.0008327945 \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), 0.0694555761, \mathsf{fma}\left(0.2909738639, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -946613766.6812615:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 670.0218683925953:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(2, 0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \mathsf{fma}\left(0.0140005442, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.0008327945 \cdot \left(x \cdot x\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0694555761, \mathsf{fma}\left(0.2909738639, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.7715471019, 1\right)\right)\right)\right)} \cdot \left(\left(\left(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + \left(\left(1 + \left(x \cdot x\right) \cdot 0.1049934947\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.0005064034\right) + 0.0001789971 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.15298196345929327}{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 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))