Average Error: 33.9 → 6.7
Time: 23.5s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.709949974998601353264013337791665792158 \cdot 10^{149}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.999993907961032104645172967888381740498 \cdot 10^{-290}:\\ \;\;\;\;1 \cdot \left(\frac{a}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)\\ \mathbf{elif}\;b_2 \le 4.275810729145967153655544315823419635021 \cdot 10^{95}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.709949974998601353264013337791665792158 \cdot 10^{149}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.999993907961032104645172967888381740498 \cdot 10^{-290}:\\
\;\;\;\;1 \cdot \left(\frac{a}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)\\

\mathbf{elif}\;b_2 \le 4.275810729145967153655544315823419635021 \cdot 10^{95}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{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 r41698 = b_2;
        double r41699 = -r41698;
        double r41700 = r41698 * r41698;
        double r41701 = a;
        double r41702 = c;
        double r41703 = r41701 * r41702;
        double r41704 = r41700 - r41703;
        double r41705 = sqrt(r41704);
        double r41706 = r41699 - r41705;
        double r41707 = r41706 / r41701;
        return r41707;
}
↓
double f(double a, double b_2, double c) {
        double r41708 = b_2;
        double r41709 = -3.7099499749986014e+149;
        bool r41710 = r41708 <= r41709;
        double r41711 = -0.5;
        double r41712 = c;
        double r41713 = r41712 / r41708;
        double r41714 = r41711 * r41713;
        double r41715 = 2.999993907961032e-290;
        bool r41716 = r41708 <= r41715;
        double r41717 = 1.0;
        double r41718 = a;
        double r41719 = r41718 / r41718;
        double r41720 = r41708 * r41708;
        double r41721 = r41718 * r41712;
        double r41722 = r41720 - r41721;
        double r41723 = sqrt(r41722);
        double r41724 = r41723 - r41708;
        double r41725 = r41712 / r41724;
        double r41726 = r41719 * r41725;
        double r41727 = r41717 * r41726;
        double r41728 = 4.275810729145967e+95;
        bool r41729 = r41708 <= r41728;
        double r41730 = -r41708;
        double r41731 = r41730 - r41723;
        double r41732 = r41731 / r41718;
        double r41733 = 0.5;
        double r41734 = r41733 * r41713;
        double r41735 = 2.0;
        double r41736 = r41708 / r41718;
        double r41737 = r41735 * r41736;
        double r41738 = r41734 - r41737;
        double r41739 = r41729 ? r41732 : r41738;
        double r41740 = r41716 ? r41727 : r41739;
        double r41741 = r41710 ? r41714 : r41740;
        return r41741;
}

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 < -3.7099499749986014e+149

    1. Initial program 63.5

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

    2. Taylor expanded around -inf 1.7

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

    if -3.7099499749986014e+149 < b_2 < 2.999993907961032e-290

    1. Initial program 34.9

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

    3. Applied flip--34.9

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

    4. Simplified16.5

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

    5. Simplified16.5

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

    7. Applied sub-neg16.5

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

    9. Applied *-un-lft-identity16.5

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

    10. Applied *-un-lft-identity16.5

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

    11. Applied *-un-lft-identity16.5

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

    12. Applied times-frac16.5

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

    13. Applied times-frac16.5

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

    14. Simplified16.5

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

    15. Simplified8.5

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

    if 2.999993907961032e-290 < b_2 < 4.275810729145967e+95

    1. Initial program 9.1

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

    if 4.275810729145967e+95 < b_2

    1. Initial program 44.3

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

    2. Taylor expanded around inf 3.6

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.709949974998601353264013337791665792158 \cdot 10^{149}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.999993907961032104645172967888381740498 \cdot 10^{-290}:\\ \;\;\;\;1 \cdot \left(\frac{a}{a} \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\right)\\ \mathbf{elif}\;b_2 \le 4.275810729145967153655544315823419635021 \cdot 10^{95}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))