Average Error: 34.3 → 8.3
Time: 20.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.359743697059274779899139718884898592231 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.608116455888013073415978967747010549618 \cdot 10^{-192}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 2.433359793914946901099874824645322603288 \cdot 10^{86}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b_2}}{2} - \frac{b_2 \cdot 2}{a}\\ \end{array}\]
\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}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r923031 = b_2;
        double r923032 = -r923031;
        double r923033 = r923031 * r923031;
        double r923034 = a;
        double r923035 = c;
        double r923036 = r923034 * r923035;
        double r923037 = r923033 - r923036;
        double r923038 = sqrt(r923037);
        double r923039 = r923032 - r923038;
        double r923040 = r923039 / r923034;
        return r923040;
}
↓
double f(double a, double b_2, double c) {
        double r923041 = b_2;
        double r923042 = -1.3597436970592748e+52;
        bool r923043 = r923041 <= r923042;
        double r923044 = -0.5;
        double r923045 = c;
        double r923046 = r923045 / r923041;
        double r923047 = r923044 * r923046;
        double r923048 = -1.608116455888013e-192;
        bool r923049 = r923041 <= r923048;
        double r923050 = a;
        double r923051 = r923041 * r923041;
        double r923052 = r923045 * r923050;
        double r923053 = r923051 - r923052;
        double r923054 = sqrt(r923053);
        double r923055 = r923054 - r923041;
        double r923056 = r923045 / r923055;
        double r923057 = r923050 * r923056;
        double r923058 = r923057 / r923050;
        double r923059 = 2.433359793914947e+86;
        bool r923060 = r923041 <= r923059;
        double r923061 = -r923041;
        double r923062 = r923061 / r923050;
        double r923063 = r923054 / r923050;
        double r923064 = r923062 - r923063;
        double r923065 = 2.0;
        double r923066 = r923046 / r923065;
        double r923067 = r923041 * r923065;
        double r923068 = r923067 / r923050;
        double r923069 = r923066 - r923068;
        double r923070 = r923060 ? r923064 : r923069;
        double r923071 = r923049 ? r923058 : r923070;
        double r923072 = r923043 ? r923047 : r923071;
        return r923072;
}

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. Taylor expanded around inf 3.9

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

    3. Simplified4.0

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

  4. Final simplification8.3

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

Reproduce

herbie shell --seed 2019170 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))