Average Error: 34.0 → 8.5
Time: 24.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.260570947330360464594776218624123716053 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.264171759716637915376042096528155176919 \cdot 10^{-294}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.092877965036258569027964750097428917598 \cdot 10^{104}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.260570947330360464594776218624123716053 \cdot 10^{-14}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.264171759716637915376042096528155176919 \cdot 10^{-294}:\\
\;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 1.092877965036258569027964750097428917598 \cdot 10^{104}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r2587976 = b_2;
        double r2587977 = -r2587976;
        double r2587978 = r2587976 * r2587976;
        double r2587979 = a;
        double r2587980 = c;
        double r2587981 = r2587979 * r2587980;
        double r2587982 = r2587978 - r2587981;
        double r2587983 = sqrt(r2587982);
        double r2587984 = r2587977 - r2587983;
        double r2587985 = r2587984 / r2587979;
        return r2587985;
}
↓
double f(double a, double b_2, double c) {
        double r2587986 = b_2;
        double r2587987 = -5.2605709473303605e-14;
        bool r2587988 = r2587986 <= r2587987;
        double r2587989 = -0.5;
        double r2587990 = c;
        double r2587991 = r2587990 / r2587986;
        double r2587992 = r2587989 * r2587991;
        double r2587993 = -3.264171759716638e-294;
        bool r2587994 = r2587986 <= r2587993;
        double r2587995 = a;
        double r2587996 = r2587986 * r2587986;
        double r2587997 = r2587995 * r2587990;
        double r2587998 = r2587996 - r2587997;
        double r2587999 = sqrt(r2587998);
        double r2588000 = r2587999 - r2587986;
        double r2588001 = r2587990 / r2588000;
        double r2588002 = r2587995 * r2588001;
        double r2588003 = r2588002 / r2587995;
        double r2588004 = 1.0928779650362586e+104;
        bool r2588005 = r2587986 <= r2588004;
        double r2588006 = -r2587986;
        double r2588007 = r2588006 - r2587999;
        double r2588008 = r2588007 / r2587995;
        double r2588009 = 0.5;
        double r2588010 = r2588009 * r2587991;
        double r2588011 = 2.0;
        double r2588012 = r2587986 / r2587995;
        double r2588013 = r2588011 * r2588012;
        double r2588014 = r2588010 - r2588013;
        double r2588015 = r2588005 ? r2588008 : r2588014;
        double r2588016 = r2587994 ? r2588003 : r2588015;
        double r2588017 = r2587988 ? r2587992 : r2588016;
        return r2588017;
}

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 < -5.2605709473303605e-14

    1. Initial program 55.9

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

    2. Taylor expanded around -inf 6.0

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

    if -5.2605709473303605e-14 < b_2 < -3.264171759716638e-294

    1. Initial program 26.5

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

    3. Applied flip--26.5

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

    4. Simplified18.3

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

    5. Simplified18.3

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

    7. Applied *-un-lft-identity18.3

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

    8. Applied times-frac14.6

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

    9. Simplified14.6

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

    if -3.264171759716638e-294 < b_2 < 1.0928779650362586e+104

    1. Initial program 9.6

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

    if 1.0928779650362586e+104 < b_2

    1. Initial program 47.7

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

    2. Taylor expanded around inf 3.3

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.260570947330360464594776218624123716053 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.264171759716637915376042096528155176919 \cdot 10^{-294}:\\ \;\;\;\;\frac{a \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 1.092877965036258569027964750097428917598 \cdot 10^{104}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))