Average Error: 33.6 → 6.9
Time: 20.7s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.1942109607748042 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.092790018801933 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 3.4243727332802643 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a \cdot \frac{1}{2}}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.1942109607748042 \cdot 10^{+147}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.092790018801933 \cdot 10^{-307}:\\
\;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\

\mathbf{elif}\;b_2 \le 3.4243727332802643 \cdot 10^{+80}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a \cdot \frac{1}{2}}\\

\end{array}
double f(double a, double b_2, double c) {
        double r861980 = b_2;
        double r861981 = -r861980;
        double r861982 = r861980 * r861980;
        double r861983 = a;
        double r861984 = c;
        double r861985 = r861983 * r861984;
        double r861986 = r861982 - r861985;
        double r861987 = sqrt(r861986);
        double r861988 = r861981 - r861987;
        double r861989 = r861988 / r861983;
        return r861989;
}
↓
double f(double a, double b_2, double c) {
        double r861990 = b_2;
        double r861991 = -1.1942109607748042e+147;
        bool r861992 = r861990 <= r861991;
        double r861993 = -0.5;
        double r861994 = c;
        double r861995 = r861994 / r861990;
        double r861996 = r861993 * r861995;
        double r861997 = -1.092790018801933e-307;
        bool r861998 = r861990 <= r861997;
        double r861999 = 1.0;
        double r862000 = r861990 * r861990;
        double r862001 = a;
        double r862002 = r862001 * r861994;
        double r862003 = r862000 - r862002;
        double r862004 = sqrt(r862003);
        double r862005 = r862004 - r861990;
        double r862006 = r862005 / r861994;
        double r862007 = r861999 / r862006;
        double r862008 = 3.4243727332802643e+80;
        bool r862009 = r861990 <= r862008;
        double r862010 = -r861990;
        double r862011 = r862010 - r862004;
        double r862012 = r861999 / r862001;
        double r862013 = r862011 * r862012;
        double r862014 = 0.5;
        double r862015 = r862014 * r861995;
        double r862016 = r862001 * r862014;
        double r862017 = r861990 / r862016;
        double r862018 = r862015 - r862017;
        double r862019 = r862009 ? r862013 : r862018;
        double r862020 = r861998 ? r862007 : r862019;
        double r862021 = r861992 ? r861996 : r862020;
        return r862021;
}

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 < -1.1942109607748042e+147

    1. Initial program 62.2

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

    2. Taylor expanded around -inf 1.3

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

    if -1.1942109607748042e+147 < b_2 < -1.092790018801933e-307

    1. Initial program 34.1

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

    3. Applied flip--34.2

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

    4. Simplified15.8

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

    5. Simplified15.8

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

    7. Applied clear-num16.0

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

    8. Simplified8.7

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

    if -1.092790018801933e-307 < b_2 < 3.4243727332802643e+80

    1. Initial program 10.0

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

    3. Applied div-inv10.1

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

    if 3.4243727332802643e+80 < b_2

    1. Initial program 41.9

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a \cdot \frac{1}{2}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.1942109607748042 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.092790018801933 \cdot 10^{-307}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{c}}\\ \mathbf{elif}\;b_2 \le 3.4243727332802643 \cdot 10^{+80}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a \cdot \frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))