Average Error: 33.2 → 6.6
Time: 18.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.364694627550853 \cdot 10^{+89}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.5510446836147797 \cdot 10^{-300}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.364694627550853 \cdot 10^{+89}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.5510446836147797 \cdot 10^{-300}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\mathbf{elif}\;b_2 \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\end{array}
double f(double a, double b_2, double c) {
        double r419661 = b_2;
        double r419662 = -r419661;
        double r419663 = r419661 * r419661;
        double r419664 = a;
        double r419665 = c;
        double r419666 = r419664 * r419665;
        double r419667 = r419663 - r419666;
        double r419668 = sqrt(r419667);
        double r419669 = r419662 - r419668;
        double r419670 = r419669 / r419664;
        return r419670;
}
↓
double f(double a, double b_2, double c) {
        double r419671 = b_2;
        double r419672 = -3.364694627550853e+89;
        bool r419673 = r419671 <= r419672;
        double r419674 = -0.5;
        double r419675 = c;
        double r419676 = r419675 / r419671;
        double r419677 = r419674 * r419676;
        double r419678 = -1.5510446836147797e-300;
        bool r419679 = r419671 <= r419678;
        double r419680 = r419671 * r419671;
        double r419681 = a;
        double r419682 = r419675 * r419681;
        double r419683 = r419680 - r419682;
        double r419684 = sqrt(r419683);
        double r419685 = r419684 - r419671;
        double r419686 = r419675 / r419685;
        double r419687 = 2.559678284282607e+69;
        bool r419688 = r419671 <= r419687;
        double r419689 = -r419671;
        double r419690 = r419689 - r419684;
        double r419691 = r419690 / r419681;
        double r419692 = 0.5;
        double r419693 = r419692 * r419676;
        double r419694 = r419671 / r419681;
        double r419695 = 2.0;
        double r419696 = r419694 * r419695;
        double r419697 = r419693 - r419696;
        double r419698 = r419688 ? r419691 : r419697;
        double r419699 = r419679 ? r419686 : r419698;
        double r419700 = r419673 ? r419677 : r419699;
        return r419700;
}

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 < -3.364694627550853e+89

    1. Initial program 57.9

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

    2. Taylor expanded around -inf 2.7

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

    if -3.364694627550853e+89 < b_2 < -1.5510446836147797e-300

    1. Initial program 31.6

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

    3. Applied flip--31.7

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

    4. Simplified15.5

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

    5. Simplified15.5

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

    7. Applied *-un-lft-identity15.5

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

    8. Applied associate-/l*15.6

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

    9. Simplified14.9

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

    11. Applied clear-num14.9

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

    12. Simplified8.7

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

    14. Applied associate-/r*8.5

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

    15. Simplified8.4

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

    if -1.5510446836147797e-300 < b_2 < 2.559678284282607e+69

    1. Initial program 9.4

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

    2. Taylor expanded around 0 9.4

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified9.4

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

    if 2.559678284282607e+69 < b_2

    1. Initial program 38.9

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

    2. Taylor expanded around inf 4.7

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.364694627550853 \cdot 10^{+89}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.5510446836147797 \cdot 10^{-300}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))