Average Error: 33.8 → 6.6
Time: 20.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.6218828469248316 \cdot 10^{82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3959529492205615 \cdot 10^{-308}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.2307425160882177 \cdot 10^{80}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.6218828469248316 \cdot 10^{82}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.3959529492205615 \cdot 10^{-308}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 1.2307425160882177 \cdot 10^{80}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r22451 = b_2;
        double r22452 = -r22451;
        double r22453 = r22451 * r22451;
        double r22454 = a;
        double r22455 = c;
        double r22456 = r22454 * r22455;
        double r22457 = r22453 - r22456;
        double r22458 = sqrt(r22457);
        double r22459 = r22452 - r22458;
        double r22460 = r22459 / r22454;
        return r22460;
}
↓
double f(double a, double b_2, double c) {
        double r22461 = b_2;
        double r22462 = -4.6218828469248316e+82;
        bool r22463 = r22461 <= r22462;
        double r22464 = -0.5;
        double r22465 = c;
        double r22466 = r22465 / r22461;
        double r22467 = r22464 * r22466;
        double r22468 = 1.3959529492205615e-308;
        bool r22469 = r22461 <= r22468;
        double r22470 = -r22461;
        double r22471 = r22461 * r22461;
        double r22472 = a;
        double r22473 = r22472 * r22465;
        double r22474 = r22471 - r22473;
        double r22475 = sqrt(r22474);
        double r22476 = r22470 + r22475;
        double r22477 = r22465 / r22476;
        double r22478 = 1.2307425160882177e+80;
        bool r22479 = r22461 <= r22478;
        double r22480 = r22470 - r22475;
        double r22481 = 1.0;
        double r22482 = r22481 / r22472;
        double r22483 = r22480 * r22482;
        double r22484 = -2.0;
        double r22485 = r22461 / r22472;
        double r22486 = r22484 * r22485;
        double r22487 = r22479 ? r22483 : r22486;
        double r22488 = r22469 ? r22477 : r22487;
        double r22489 = r22463 ? r22467 : r22488;
        return r22489;
}

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 < -4.6218828469248316e+82

    1. Initial program 58.2

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

    2. Taylor expanded around -inf 2.7

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

    if -4.6218828469248316e+82 < b_2 < 1.3959529492205615e-308

    1. Initial program 30.4

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

    3. Applied div-inv30.4

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

    5. Applied flip--30.4

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

    6. Applied associate-*l/30.5

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

    7. Simplified16.1

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

    9. Applied *-un-lft-identity16.1

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

    10. Applied *-un-lft-identity16.1

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

    11. Applied times-frac16.1

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

    12. Simplified16.1

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

    13. Simplified9.2

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

    if 1.3959529492205615e-308 < b_2 < 1.2307425160882177e+80

    1. Initial program 8.4

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

    3. Applied div-inv8.6

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

    if 1.2307425160882177e+80 < b_2

    1. Initial program 44.4

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

    3. Applied clear-num44.4

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

    4. Taylor expanded around 0 4.5

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.6218828469248316 \cdot 10^{82}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.3959529492205615 \cdot 10^{-308}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 1.2307425160882177 \cdot 10^{80}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))