Average Error: 33.0 → 10.8
Time: 16.8s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.782434836873191 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.990519652731023 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.0350377446088803 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.325219738594455 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2} \cdot \frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 + c \cdot a\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.782434836873191 \cdot 10^{+41}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le 1.990519652731023 \cdot 10^{-106}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.0350377446088803 \cdot 10^{-69}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.325219738594455 \cdot 10^{-21}:\\
\;\;\;\;\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2} \cdot \frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 + c \cdot a\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r872700 = b_2;
        double r872701 = -r872700;
        double r872702 = r872700 * r872700;
        double r872703 = a;
        double r872704 = c;
        double r872705 = r872703 * r872704;
        double r872706 = r872702 - r872705;
        double r872707 = sqrt(r872706);
        double r872708 = r872701 + r872707;
        double r872709 = r872708 / r872703;
        return r872709;
}
↓
double f(double a, double b_2, double c) {
        double r872710 = b_2;
        double r872711 = -2.782434836873191e+41;
        bool r872712 = r872710 <= r872711;
        double r872713 = 0.5;
        double r872714 = c;
        double r872715 = r872714 / r872710;
        double r872716 = r872713 * r872715;
        double r872717 = a;
        double r872718 = r872710 / r872717;
        double r872719 = 2.0;
        double r872720 = r872718 * r872719;
        double r872721 = r872716 - r872720;
        double r872722 = 1.990519652731023e-106;
        bool r872723 = r872710 <= r872722;
        double r872724 = 1.0;
        double r872725 = r872710 * r872710;
        double r872726 = r872714 * r872717;
        double r872727 = r872725 - r872726;
        double r872728 = sqrt(r872727);
        double r872729 = r872728 - r872710;
        double r872730 = r872717 / r872729;
        double r872731 = r872724 / r872730;
        double r872732 = 1.0350377446088803e-69;
        bool r872733 = r872710 <= r872732;
        double r872734 = -0.5;
        double r872735 = r872734 * r872715;
        double r872736 = 3.325219738594455e-21;
        bool r872737 = r872710 <= r872736;
        double r872738 = r872728 + r872710;
        double r872739 = r872724 / r872738;
        double r872740 = r872725 + r872726;
        double r872741 = r872725 - r872740;
        double r872742 = r872741 / r872717;
        double r872743 = r872739 * r872742;
        double r872744 = r872737 ? r872743 : r872735;
        double r872745 = r872733 ? r872735 : r872744;
        double r872746 = r872723 ? r872731 : r872745;
        double r872747 = r872712 ? r872721 : r872746;
        return r872747;
}

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 < -2.782434836873191e+41

    1. Initial program 34.1

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

    2. Simplified34.1

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

    3. Taylor expanded around -inf 6.3

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

    if -2.782434836873191e+41 < b_2 < 1.990519652731023e-106

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied clear-num12.9

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

    if 1.990519652731023e-106 < b_2 < 1.0350377446088803e-69 or 3.325219738594455e-21 < b_2

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 8.7

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

    if 1.0350377446088803e-69 < b_2 < 3.325219738594455e-21

    1. Initial program 35.5

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

    2. Simplified35.5

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

    4. Applied clear-num35.5

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\]
    5. Using strategy rm

    6. Applied flip--35.6

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

    7. Applied associate-/r/35.7

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

    8. Applied add-cube-cbrt35.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2\right)}\]

    9. Applied times-frac35.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}} \cdot \frac{\sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}\]

    10. Simplified35.6

      \[\leadsto \color{blue}{\frac{b_2 \cdot b_2 - \left(a \cdot c + b_2 \cdot b_2\right)}{a}} \cdot \frac{\sqrt[3]{1}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\]

    11. Simplified35.6

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.782434836873191 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.990519652731023 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.0350377446088803 \cdot 10^{-69}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.325219738594455 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2} \cdot \frac{b_2 \cdot b_2 - \left(b_2 \cdot b_2 + c \cdot a\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))