Average Error: 34.2 → 7.1
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.501658503658959535381027650257422342525 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.203463322495104127662247988126463798444 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.501658503658959535381027650257422342525 \cdot 10^{153}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.203463322495104127662247988126463798444 \cdot 10^{-228}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\
\;\;\;\;\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} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r14208 = b_2;
        double r14209 = -r14208;
        double r14210 = r14208 * r14208;
        double r14211 = a;
        double r14212 = c;
        double r14213 = r14211 * r14212;
        double r14214 = r14210 - r14213;
        double r14215 = sqrt(r14214);
        double r14216 = r14209 - r14215;
        double r14217 = r14216 / r14211;
        return r14217;
}
↓
double f(double a, double b_2, double c) {
        double r14218 = b_2;
        double r14219 = -9.50165850365896e+153;
        bool r14220 = r14218 <= r14219;
        double r14221 = -0.5;
        double r14222 = c;
        double r14223 = r14222 / r14218;
        double r14224 = r14221 * r14223;
        double r14225 = 3.203463322495104e-228;
        bool r14226 = r14218 <= r14225;
        double r14227 = 1.0;
        double r14228 = a;
        double r14229 = r14227 / r14228;
        double r14230 = 0.0;
        double r14231 = r14229 * r14230;
        double r14232 = r14227 * r14222;
        double r14233 = r14231 + r14232;
        double r14234 = -r14218;
        double r14235 = r14218 * r14218;
        double r14236 = r14228 * r14222;
        double r14237 = r14235 - r14236;
        double r14238 = sqrt(r14237);
        double r14239 = r14234 + r14238;
        double r14240 = r14233 / r14239;
        double r14241 = 6.568668442325333e+48;
        bool r14242 = r14218 <= r14241;
        double r14243 = r14234 - r14238;
        double r14244 = r14243 * r14229;
        double r14245 = 0.5;
        double r14246 = r14245 * r14223;
        double r14247 = 2.0;
        double r14248 = r14218 / r14228;
        double r14249 = r14247 * r14248;
        double r14250 = r14246 - r14249;
        double r14251 = r14242 ? r14244 : r14250;
        double r14252 = r14226 ? r14240 : r14251;
        double r14253 = r14220 ? r14224 : r14252;
        return r14253;
}

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 < -9.50165850365896e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 1.6

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

    if -9.50165850365896e+153 < b_2 < 3.203463322495104e-228

    1. Initial program 31.4

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

    3. Applied div-inv31.5

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

    5. Applied flip--31.5

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

    6. Applied associate-*l/31.6

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

    7. Simplified14.7

      \[\leadsto \frac{\color{blue}{\frac{1}{a} \cdot 0 + \frac{1}{a} \cdot \left(a \cdot c\right)}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
    8. Using strategy rm

    9. Applied associate-*r*9.3

      \[\leadsto \frac{\frac{1}{a} \cdot 0 + \color{blue}{\left(\frac{1}{a} \cdot a\right) \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    10. Simplified9.2

      \[\leadsto \frac{\frac{1}{a} \cdot 0 + \color{blue}{1} \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    if 3.203463322495104e-228 < b_2 < 6.568668442325333e+48

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 6.568668442325333e+48 < b_2

    1. Initial program 39.1

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

    2. Taylor expanded around inf 5.3

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.501658503658959535381027650257422342525 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.203463322495104127662247988126463798444 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))