Average Error: 29.0 → 0.3
Time: 1.1m
Precision: 64
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.1349577297696527:\\ \;\;\;\;\frac{0.15298196345929327}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 1.1391870338076322:\\ \;\;\;\;\left({x}^{5} \cdot 0.265709700396151 + x\right) - 0.6665536072 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.15298196345929327}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\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 -1.1349577297696527:\\
\;\;\;\;\frac{0.15298196345929327}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 1.1391870338076322:\\
\;\;\;\;\left({x}^{5} \cdot 0.265709700396151 + x\right) - 0.6665536072 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{0.15298196345929327}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)}\right)\\

\end{array}
double f(double x) {
        double r8344552 = 1.0;
        double r8344553 = 0.1049934947;
        double r8344554 = x;
        double r8344555 = r8344554 * r8344554;
        double r8344556 = r8344553 * r8344555;
        double r8344557 = r8344552 + r8344556;
        double r8344558 = 0.0424060604;
        double r8344559 = r8344555 * r8344555;
        double r8344560 = r8344558 * r8344559;
        double r8344561 = r8344557 + r8344560;
        double r8344562 = 0.0072644182;
        double r8344563 = r8344559 * r8344555;
        double r8344564 = r8344562 * r8344563;
        double r8344565 = r8344561 + r8344564;
        double r8344566 = 0.0005064034;
        double r8344567 = r8344563 * r8344555;
        double r8344568 = r8344566 * r8344567;
        double r8344569 = r8344565 + r8344568;
        double r8344570 = 0.0001789971;
        double r8344571 = r8344567 * r8344555;
        double r8344572 = r8344570 * r8344571;
        double r8344573 = r8344569 + r8344572;
        double r8344574 = 0.7715471019;
        double r8344575 = r8344574 * r8344555;
        double r8344576 = r8344552 + r8344575;
        double r8344577 = 0.2909738639;
        double r8344578 = r8344577 * r8344559;
        double r8344579 = r8344576 + r8344578;
        double r8344580 = 0.0694555761;
        double r8344581 = r8344580 * r8344563;
        double r8344582 = r8344579 + r8344581;
        double r8344583 = 0.0140005442;
        double r8344584 = r8344583 * r8344567;
        double r8344585 = r8344582 + r8344584;
        double r8344586 = 0.0008327945;
        double r8344587 = r8344586 * r8344571;
        double r8344588 = r8344585 + r8344587;
        double r8344589 = 2.0;
        double r8344590 = r8344589 * r8344570;
        double r8344591 = r8344571 * r8344555;
        double r8344592 = r8344590 * r8344591;
        double r8344593 = r8344588 + r8344592;
        double r8344594 = r8344573 / r8344593;
        double r8344595 = r8344594 * r8344554;
        return r8344595;
}
↓
double f(double x) {
        double r8344596 = x;
        double r8344597 = -1.1349577297696527;
        bool r8344598 = r8344596 <= r8344597;
        double r8344599 = 0.15298196345929327;
        double r8344600 = 5.0;
        double r8344601 = pow(r8344596, r8344600);
        double r8344602 = r8344599 / r8344601;
        double r8344603 = 0.5;
        double r8344604 = r8344603 / r8344596;
        double r8344605 = 0.2514179000665375;
        double r8344606 = r8344596 * r8344596;
        double r8344607 = r8344596 * r8344606;
        double r8344608 = r8344605 / r8344607;
        double r8344609 = r8344604 + r8344608;
        double r8344610 = r8344602 + r8344609;
        double r8344611 = 1.1391870338076322;
        bool r8344612 = r8344596 <= r8344611;
        double r8344613 = 0.265709700396151;
        double r8344614 = r8344601 * r8344613;
        double r8344615 = r8344614 + r8344596;
        double r8344616 = 0.6665536072;
        double r8344617 = r8344616 * r8344607;
        double r8344618 = r8344615 - r8344617;
        double r8344619 = r8344612 ? r8344618 : r8344610;
        double r8344620 = r8344598 ? r8344610 : r8344619;
        return r8344620;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -1.1349577297696527 or 1.1391870338076322 < x

    1. Initial program 57.7

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

    if -1.1349577297696527 < x < 1.1391870338076322

    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. Simplified0.0

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

    3. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{\left(x + 0.265709700396151 \cdot {x}^{5}\right) - 0.6665536072 \cdot {x}^{3}}\]

    4. Simplified0.3

      \[\leadsto \color{blue}{\left({x}^{5} \cdot 0.265709700396151 + x\right) - 0.6665536072 \cdot \left(x \cdot \left(x \cdot x\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.1349577297696527:\\ \;\;\;\;\frac{0.15298196345929327}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 1.1391870338076322:\\ \;\;\;\;\left({x}^{5} \cdot 0.265709700396151 + x\right) - 0.6665536072 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.15298196345929327}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))