Average Error: 34.6 → 6.5
Time: 24.2s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le -3.3528823044057167 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.2718773002973155 \cdot 10^{78}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.6581383089037873 \cdot 10^{81}:\\
\;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \le -3.3528823044057167 \cdot 10^{-206}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.2718773002973155 \cdot 10^{78}:\\
\;\;\;\;\frac{c}{\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 r28705 = b_2;
        double r28706 = -r28705;
        double r28707 = r28705 * r28705;
        double r28708 = a;
        double r28709 = c;
        double r28710 = r28708 * r28709;
        double r28711 = r28707 - r28710;
        double r28712 = sqrt(r28711);
        double r28713 = r28706 + r28712;
        double r28714 = r28713 / r28708;
        return r28714;
}
↓
double f(double a, double b_2, double c) {
        double r28715 = b_2;
        double r28716 = -1.6581383089037873e+81;
        bool r28717 = r28715 <= r28716;
        double r28718 = c;
        double r28719 = r28718 / r28715;
        double r28720 = 0.5;
        double r28721 = a;
        double r28722 = r28715 / r28721;
        double r28723 = -2.0;
        double r28724 = r28722 * r28723;
        double r28725 = fma(r28719, r28720, r28724);
        double r28726 = -3.3528823044057167e-206;
        bool r28727 = r28715 <= r28726;
        double r28728 = 1.0;
        double r28729 = r28715 * r28715;
        double r28730 = r28721 * r28718;
        double r28731 = r28729 - r28730;
        double r28732 = sqrt(r28731);
        double r28733 = r28732 - r28715;
        double r28734 = r28721 / r28733;
        double r28735 = r28728 / r28734;
        double r28736 = 1.2718773002973155e+78;
        bool r28737 = r28715 <= r28736;
        double r28738 = -r28715;
        double r28739 = r28738 - r28732;
        double r28740 = r28718 / r28739;
        double r28741 = -0.5;
        double r28742 = r28741 * r28719;
        double r28743 = r28737 ? r28740 : r28742;
        double r28744 = r28727 ? r28735 : r28743;
        double r28745 = r28717 ? r28725 : r28744;
        return r28745;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.6581383089037873e+81

    1. Initial program 43.9

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

    2. Taylor expanded around -inf 3.7

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

    3. Simplified3.7

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

    if -1.6581383089037873e+81 < b_2 < -3.3528823044057167e-206

    1. Initial program 7.7

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

    3. Applied clear-num7.9

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

    4. Simplified7.9

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

    if -3.3528823044057167e-206 < b_2 < 1.2718773002973155e+78

    1. Initial program 28.9

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

    3. Applied flip-+29.1

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

    4. Simplified16.6

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

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

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

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

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

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

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

    9. Applied times-frac16.6

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

    10. Applied times-frac16.6

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

    11. Simplified16.6

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

    12. Simplified22.2

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

    14. Applied associate-/r*15.9

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

    15. Simplified9.7

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

    if 1.2718773002973155e+78 < b_2

    1. Initial program 59.0

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.6581383089037873 \cdot 10^{81}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le -3.3528823044057167 \cdot 10^{-206}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.2718773002973155 \cdot 10^{78}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))