Average Error: 33.8 → 10.2
Time: 21.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r954158 = b_2;
        double r954159 = -r954158;
        double r954160 = r954158 * r954158;
        double r954161 = a;
        double r954162 = c;
        double r954163 = r954161 * r954162;
        double r954164 = r954160 - r954163;
        double r954165 = sqrt(r954164);
        double r954166 = r954159 - r954165;
        double r954167 = r954166 / r954161;
        return r954167;
}
↓
double f(double a, double b_2, double c) {
        double r954168 = b_2;
        double r954169 = -2.5694949190681246e-64;
        bool r954170 = r954168 <= r954169;
        double r954171 = -0.5;
        double r954172 = c;
        double r954173 = r954172 / r954168;
        double r954174 = r954171 * r954173;
        double r954175 = 2.865381670376961e+117;
        bool r954176 = r954168 <= r954175;
        double r954177 = -r954168;
        double r954178 = r954168 * r954168;
        double r954179 = a;
        double r954180 = r954179 * r954172;
        double r954181 = r954178 - r954180;
        double r954182 = sqrt(r954181);
        double r954183 = r954177 - r954182;
        double r954184 = r954183 / r954179;
        double r954185 = r954168 / r954179;
        double r954186 = -2.0;
        double r954187 = r954185 * r954186;
        double r954188 = r954176 ? r954184 : r954187;
        double r954189 = r954170 ? r954174 : r954188;
        return r954189;
}

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 3 regimes

  2. if b_2 < -2.5694949190681246e-64

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 9.1

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

    if -2.5694949190681246e-64 < b_2 < 2.865381670376961e+117

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    5. Applied un-div-inv13.1

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

    if 2.865381670376961e+117 < b_2

    1. Initial program 52.0

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

    3. Applied clear-num52.1

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

    4. Taylor expanded around 0 3.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))