Average Error: 33.9 → 10.3
Time: 21.3s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.664666030704839304934680918144560500037 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 5.713258900191085490618735432194689062256 \cdot 10^{74}:\\ \;\;\;\;c \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -7.664666030704839304934680918144560500037 \cdot 10^{-132}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le 5.713258900191085490618735432194689062256 \cdot 10^{74}:\\
\;\;\;\;c \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r28686 = b_2;
        double r28687 = -r28686;
        double r28688 = r28686 * r28686;
        double r28689 = a;
        double r28690 = c;
        double r28691 = r28689 * r28690;
        double r28692 = r28688 - r28691;
        double r28693 = sqrt(r28692);
        double r28694 = r28687 + r28693;
        double r28695 = r28694 / r28689;
        return r28695;
}
↓
double f(double a, double b_2, double c) {
        double r28696 = b_2;
        double r28697 = -7.664666030704839e-132;
        bool r28698 = r28696 <= r28697;
        double r28699 = c;
        double r28700 = r28699 / r28696;
        double r28701 = 0.5;
        double r28702 = a;
        double r28703 = r28696 / r28702;
        double r28704 = -2.0;
        double r28705 = r28703 * r28704;
        double r28706 = fma(r28700, r28701, r28705);
        double r28707 = 5.713258900191085e+74;
        bool r28708 = r28696 <= r28707;
        double r28709 = 1.0;
        double r28710 = -r28696;
        double r28711 = r28696 * r28696;
        double r28712 = r28702 * r28699;
        double r28713 = r28711 - r28712;
        double r28714 = sqrt(r28713);
        double r28715 = r28710 - r28714;
        double r28716 = r28709 / r28715;
        double r28717 = r28699 * r28716;
        double r28718 = -0.5;
        double r28719 = r28718 * r28700;
        double r28720 = r28708 ? r28717 : r28719;
        double r28721 = r28698 ? r28706 : r28720;
        return r28721;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -7.664666030704839e-132

    1. Initial program 24.4

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

    2. Taylor expanded around -inf 14.0

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

    3. Simplified14.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)}\]

    if -7.664666030704839e-132 < b_2 < 5.713258900191085e+74

    1. Initial program 26.2

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

    3. Applied flip-+26.9

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

    4. Simplified16.5

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

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

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

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

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

    8. Applied times-frac14.9

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

    9. Applied times-frac12.1

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

    10. Simplified12.1

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

    11. Simplified11.4

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

    if 5.713258900191085e+74 < b_2

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.664666030704839304934680918144560500037 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 5.713258900191085490618735432194689062256 \cdot 10^{74}:\\ \;\;\;\;c \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))