Average Error: 34.0 → 7.8
Time: 17.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -977083.9042033920995891094207763671875:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.419897611628610882899329113648886874763 \cdot 10^{-291}:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 6.073987741152761147577909172394710657791 \cdot 10^{104}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(-2 \cdot b_2\right)}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -977083.9042033920995891094207763671875:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.419897611628610882899329113648886874763 \cdot 10^{-291}:\\
\;\;\;\;\frac{1 \cdot \frac{a}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 6.073987741152761147577909172394710657791 \cdot 10^{104}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1 \cdot \left(-2 \cdot b_2\right)}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r25426 = b_2;
        double r25427 = -r25426;
        double r25428 = r25426 * r25426;
        double r25429 = a;
        double r25430 = c;
        double r25431 = r25429 * r25430;
        double r25432 = r25428 - r25431;
        double r25433 = sqrt(r25432);
        double r25434 = r25427 - r25433;
        double r25435 = r25434 / r25429;
        return r25435;
}
↓
double f(double a, double b_2, double c) {
        double r25436 = b_2;
        double r25437 = -977083.9042033921;
        bool r25438 = r25436 <= r25437;
        double r25439 = -0.5;
        double r25440 = c;
        double r25441 = r25440 / r25436;
        double r25442 = r25439 * r25441;
        double r25443 = -1.4198976116286109e-291;
        bool r25444 = r25436 <= r25443;
        double r25445 = 1.0;
        double r25446 = a;
        double r25447 = r25436 * r25436;
        double r25448 = r25440 * r25446;
        double r25449 = r25447 - r25448;
        double r25450 = sqrt(r25449);
        double r25451 = r25450 - r25436;
        double r25452 = r25451 / r25440;
        double r25453 = r25446 / r25452;
        double r25454 = r25445 * r25453;
        double r25455 = r25454 / r25446;
        double r25456 = 6.073987741152761e+104;
        bool r25457 = r25436 <= r25456;
        double r25458 = -r25436;
        double r25459 = r25446 * r25440;
        double r25460 = r25447 - r25459;
        double r25461 = sqrt(r25460);
        double r25462 = r25458 - r25461;
        double r25463 = r25462 / r25446;
        double r25464 = -2.0;
        double r25465 = r25464 * r25436;
        double r25466 = r25445 * r25465;
        double r25467 = r25466 / r25446;
        double r25468 = r25457 ? r25463 : r25467;
        double r25469 = r25444 ? r25455 : r25468;
        double r25470 = r25438 ? r25442 : r25469;
        return r25470;
}

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 < -977083.9042033921

    1. Initial program 56.3

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

    2. Taylor expanded around -inf 5.0

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

    if -977083.9042033921 < b_2 < -1.4198976116286109e-291

    1. Initial program 27.3

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

    3. Applied flip--27.3

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

    4. Simplified17.2

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

    5. Simplified17.2

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity17.2

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

    8. Applied *-un-lft-identity17.2

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

    9. Applied times-frac17.2

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

    10. Simplified17.2

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

    11. Simplified14.0

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

    if -1.4198976116286109e-291 < b_2 < 6.073987741152761e+104

    1. Initial program 8.6

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

    if 6.073987741152761e+104 < b_2

    1. Initial program 47.7

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

    3. Applied flip--63.4

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

    4. Simplified62.5

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

    5. Simplified62.5

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity62.5

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

    8. Applied *-un-lft-identity62.5

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

    9. Applied times-frac62.5

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

    10. Simplified62.5

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

    11. Simplified62.3

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

    12. Taylor expanded around 0 3.7

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

  4. Final simplification7.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -977083.9042033920995891094207763671875:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.419897611628610882899329113648886874763 \cdot 10^{-291}:\\ \;\;\;\;\frac{1 \cdot \frac{a}{\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 6.073987741152761147577909172394710657791 \cdot 10^{104}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(-2 \cdot b_2\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))