Average Error: 33.1 → 6.3
Time: 29.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.709051322868216 \cdot 10^{+150}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.5270348471001126 \cdot 10^{-284}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 7.774374944213099 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.709051322868216 \cdot 10^{+150}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.5270348471001126 \cdot 10^{-284}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 7.774374944213099 \cdot 10^{+79}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1212261 = b_2;
        double r1212262 = -r1212261;
        double r1212263 = r1212261 * r1212261;
        double r1212264 = a;
        double r1212265 = c;
        double r1212266 = r1212264 * r1212265;
        double r1212267 = r1212263 - r1212266;
        double r1212268 = sqrt(r1212267);
        double r1212269 = r1212262 - r1212268;
        double r1212270 = r1212269 / r1212264;
        return r1212270;
}
↓
double f(double a, double b_2, double c) {
        double r1212271 = b_2;
        double r1212272 = -5.709051322868216e+150;
        bool r1212273 = r1212271 <= r1212272;
        double r1212274 = -0.5;
        double r1212275 = c;
        double r1212276 = r1212275 / r1212271;
        double r1212277 = r1212274 * r1212276;
        double r1212278 = 1.5270348471001126e-284;
        bool r1212279 = r1212271 <= r1212278;
        double r1212280 = r1212271 * r1212271;
        double r1212281 = a;
        double r1212282 = r1212281 * r1212275;
        double r1212283 = r1212280 - r1212282;
        double r1212284 = sqrt(r1212283);
        double r1212285 = r1212284 - r1212271;
        double r1212286 = r1212275 / r1212285;
        double r1212287 = 7.774374944213099e+79;
        bool r1212288 = r1212271 <= r1212287;
        double r1212289 = 1.0;
        double r1212290 = -r1212271;
        double r1212291 = r1212290 - r1212284;
        double r1212292 = r1212281 / r1212291;
        double r1212293 = r1212289 / r1212292;
        double r1212294 = -2.0;
        double r1212295 = r1212271 / r1212281;
        double r1212296 = r1212294 * r1212295;
        double r1212297 = r1212288 ? r1212293 : r1212296;
        double r1212298 = r1212279 ? r1212286 : r1212297;
        double r1212299 = r1212273 ? r1212277 : r1212298;
        return r1212299;
}

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 < -5.709051322868216e+150

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 1.1

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

    if -5.709051322868216e+150 < b_2 < 1.5270348471001126e-284

    1. Initial program 32.2

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

    3. Applied flip--32.3

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

    4. Simplified15.3

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

    5. Simplified15.3

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

    7. Applied *-un-lft-identity15.3

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

    8. Applied *-un-lft-identity15.3

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

    9. Applied *-un-lft-identity15.3

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

    10. Applied distribute-lft-out--15.3

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

    11. Applied *-un-lft-identity15.3

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

    12. Applied times-frac15.3

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

    13. Applied times-frac15.3

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

    14. Simplified15.3

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

    15. Simplified19.2

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

    17. Applied associate-/r*13.8

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

    18. Simplified7.9

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

    if 1.5270348471001126e-284 < b_2 < 7.774374944213099e+79

    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 *-un-lft-identity9.1

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

    4. Applied associate-/l*9.3

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

    if 7.774374944213099e+79 < b_2

    1. Initial program 40.0

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

    3. Applied *-un-lft-identity40.0

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

    4. Applied associate-/l*40.1

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

    5. Taylor expanded around 0 4.2

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.709051322868216 \cdot 10^{+150}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.5270348471001126 \cdot 10^{-284}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 7.774374944213099 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +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))