Average Error: 34.2 → 10.1
Time: 5.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\
\;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\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 r18223 = b_2;
        double r18224 = -r18223;
        double r18225 = r18223 * r18223;
        double r18226 = a;
        double r18227 = c;
        double r18228 = r18226 * r18227;
        double r18229 = r18225 - r18228;
        double r18230 = sqrt(r18229);
        double r18231 = r18224 - r18230;
        double r18232 = r18231 / r18226;
        return r18232;
}
↓
double f(double a, double b_2, double c) {
        double r18233 = b_2;
        double r18234 = -4.011579732710567e-81;
        bool r18235 = r18233 <= r18234;
        double r18236 = -0.5;
        double r18237 = c;
        double r18238 = r18237 / r18233;
        double r18239 = r18236 * r18238;
        double r18240 = 1.3176462918432122e+99;
        bool r18241 = r18233 <= r18240;
        double r18242 = 1.0;
        double r18243 = a;
        double r18244 = r18242 / r18243;
        double r18245 = -r18233;
        double r18246 = r18233 * r18233;
        double r18247 = r18243 * r18237;
        double r18248 = r18246 - r18247;
        double r18249 = sqrt(r18248);
        double r18250 = r18245 - r18249;
        double r18251 = r18242 / r18250;
        double r18252 = r18244 / r18251;
        double r18253 = 0.5;
        double r18254 = r18253 * r18238;
        double r18255 = 2.0;
        double r18256 = r18233 / r18243;
        double r18257 = r18255 * r18256;
        double r18258 = r18254 - r18257;
        double r18259 = r18241 ? r18252 : r18258;
        double r18260 = r18235 ? r18239 : r18259;
        return r18260;
}

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 < -4.011579732710567e-81

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.4

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

    if -4.011579732710567e-81 < b_2 < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied clear-num13.0

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

    5. Applied div-inv13.1

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

    6. Applied associate-/r*13.1

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

    if 1.3176462918432122e+99 < b_2

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))