Average Error: 34.3 → 8.3
Time: 19.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.359743697059274779899139718884898592231 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.608116455888013073415978967747010549618 \cdot 10^{-192}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.433359793914946901099874824645322603288 \cdot 10^{86}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.359743697059274779899139718884898592231 \cdot 10^{52}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.608116455888013073415978967747010549618 \cdot 10^{-192}:\\
\;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 2.433359793914946901099874824645322603288 \cdot 10^{86}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b_2}}{2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r3403508 = b_2;
        double r3403509 = -r3403508;
        double r3403510 = r3403508 * r3403508;
        double r3403511 = a;
        double r3403512 = c;
        double r3403513 = r3403511 * r3403512;
        double r3403514 = r3403510 - r3403513;
        double r3403515 = sqrt(r3403514);
        double r3403516 = r3403509 - r3403515;
        double r3403517 = r3403516 / r3403511;
        return r3403517;
}
↓
double f(double a, double b_2, double c) {
        double r3403518 = b_2;
        double r3403519 = -1.3597436970592748e+52;
        bool r3403520 = r3403518 <= r3403519;
        double r3403521 = -0.5;
        double r3403522 = c;
        double r3403523 = r3403522 / r3403518;
        double r3403524 = r3403521 * r3403523;
        double r3403525 = -1.608116455888013e-192;
        bool r3403526 = r3403518 <= r3403525;
        double r3403527 = a;
        double r3403528 = r3403518 * r3403518;
        double r3403529 = r3403522 * r3403527;
        double r3403530 = r3403528 - r3403529;
        double r3403531 = sqrt(r3403530);
        double r3403532 = r3403531 - r3403518;
        double r3403533 = r3403522 / r3403532;
        double r3403534 = r3403527 * r3403533;
        double r3403535 = r3403534 / r3403527;
        double r3403536 = 2.433359793914947e+86;
        bool r3403537 = r3403518 <= r3403536;
        double r3403538 = r3403518 / r3403527;
        double r3403539 = -r3403538;
        double r3403540 = r3403531 / r3403527;
        double r3403541 = r3403539 - r3403540;
        double r3403542 = 2.0;
        double r3403543 = r3403523 / r3403542;
        double r3403544 = r3403542 * r3403538;
        double r3403545 = r3403543 - r3403544;
        double r3403546 = r3403537 ? r3403541 : r3403545;
        double r3403547 = r3403526 ? r3403535 : r3403546;
        double r3403548 = r3403520 ? r3403524 : r3403547;
        return r3403548;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.3597436970592748e+52

    1. Initial program 57.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 4.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.3597436970592748e+52 < b_2 < -1.608116455888013e-192

    1. Initial program 34.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--34.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified18.2

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified18.2

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity18.2

      \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}}{a}\]

    8. Applied times-frac14.7

      \[\leadsto \frac{\color{blue}{\frac{a}{1} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    9. Simplified14.7

      \[\leadsto \frac{\color{blue}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\]

    if -1.608116455888013e-192 < b_2 < 2.433359793914947e+86

    1. Initial program 10.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub10.5

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    if 2.433359793914947e+86 < b_2

    1. Initial program 44.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-sub44.0

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    4. Taylor expanded around inf 3.9

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    5. Simplified3.9

      \[\leadsto \color{blue}{\frac{\frac{c}{b_2}}{2} - \frac{b_2}{a} \cdot 2}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.359743697059274779899139718884898592231 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.608116455888013073415978967747010549618 \cdot 10^{-192}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.433359793914946901099874824645322603288 \cdot 10^{86}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))