Average Error: 33.3 → 8.5
Time: 17.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.2863876983616346 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.2374193672365033 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.340270116328134 \cdot 10^{+83}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.2863876983616346 \cdot 10^{-14}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.2374193672365033 \cdot 10^{-189}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 3.340270116328134 \cdot 10^{+83}:\\
\;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r466970 = b_2;
        double r466971 = -r466970;
        double r466972 = r466970 * r466970;
        double r466973 = a;
        double r466974 = c;
        double r466975 = r466973 * r466974;
        double r466976 = r466972 - r466975;
        double r466977 = sqrt(r466976);
        double r466978 = r466971 - r466977;
        double r466979 = r466978 / r466973;
        return r466979;
}
↓
double f(double a, double b_2, double c) {
        double r466980 = b_2;
        double r466981 = -4.2863876983616346e-14;
        bool r466982 = r466980 <= r466981;
        double r466983 = -0.5;
        double r466984 = c;
        double r466985 = r466984 / r466980;
        double r466986 = r466983 * r466985;
        double r466987 = -3.2374193672365033e-189;
        bool r466988 = r466980 <= r466987;
        double r466989 = a;
        double r466990 = r466984 * r466989;
        double r466991 = r466980 * r466980;
        double r466992 = r466991 - r466990;
        double r466993 = sqrt(r466992);
        double r466994 = r466993 - r466980;
        double r466995 = r466990 / r466994;
        double r466996 = r466995 / r466989;
        double r466997 = 3.340270116328134e+83;
        bool r466998 = r466980 <= r466997;
        double r466999 = r466993 + r466980;
        double r467000 = r466999 / r466989;
        double r467001 = -r467000;
        double r467002 = 0.5;
        double r467003 = r467002 * r466985;
        double r467004 = r466980 / r466989;
        double r467005 = 2.0;
        double r467006 = r467004 * r467005;
        double r467007 = r467003 - r467006;
        double r467008 = r466998 ? r467001 : r467007;
        double r467009 = r466988 ? r466996 : r467008;
        double r467010 = r466982 ? r466986 : r467009;
        return r467010;
}

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 < -4.2863876983616346e-14

    1. Initial program 54.8

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

    2. Taylor expanded around -inf 5.9

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

    if -4.2863876983616346e-14 < b_2 < -3.2374193672365033e-189

    1. Initial program 29.3

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

    3. Applied flip--29.4

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

    4. Simplified18.0

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified18.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    if -3.2374193672365033e-189 < b_2 < 3.340270116328134e+83

    1. Initial program 9.8

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

    3. Applied *-un-lft-identity9.8

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

    4. Applied associate-/r*9.8

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

    5. Simplified9.8

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

    if 3.340270116328134e+83 < b_2

    1. Initial program 42.6

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

    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. Recombined 4 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.2863876983616346 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.2374193672365033 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 3.340270116328134 \cdot 10^{+83}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))