Average Error: 34.2 → 6.8
Time: 13.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.766148915908884171942137020871314382353 \cdot 10^{131}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.981443107382582658022271728267487907499 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 3.310436447834858066850098837567764712113 \cdot 10^{94}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.766148915908884171942137020871314382353 \cdot 10^{131}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -2.981443107382582658022271728267487907499 \cdot 10^{-294}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 3.310436447834858066850098837567764712113 \cdot 10^{94}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot 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 r22722 = b_2;
        double r22723 = -r22722;
        double r22724 = r22722 * r22722;
        double r22725 = a;
        double r22726 = c;
        double r22727 = r22725 * r22726;
        double r22728 = r22724 - r22727;
        double r22729 = sqrt(r22728);
        double r22730 = r22723 + r22729;
        double r22731 = r22730 / r22725;
        return r22731;
}
↓
double f(double a, double b_2, double c) {
        double r22732 = b_2;
        double r22733 = -3.766148915908884e+131;
        bool r22734 = r22732 <= r22733;
        double r22735 = 0.5;
        double r22736 = c;
        double r22737 = r22736 / r22732;
        double r22738 = r22735 * r22737;
        double r22739 = 2.0;
        double r22740 = a;
        double r22741 = r22732 / r22740;
        double r22742 = r22739 * r22741;
        double r22743 = r22738 - r22742;
        double r22744 = -2.9814431073825827e-294;
        bool r22745 = r22732 <= r22744;
        double r22746 = 1.0;
        double r22747 = r22732 * r22732;
        double r22748 = r22740 * r22736;
        double r22749 = r22747 - r22748;
        double r22750 = sqrt(r22749);
        double r22751 = r22750 - r22732;
        double r22752 = r22740 / r22751;
        double r22753 = r22746 / r22752;
        double r22754 = 3.310436447834858e+94;
        bool r22755 = r22732 <= r22754;
        double r22756 = r22746 / r22740;
        double r22757 = 0.0;
        double r22758 = r22756 * r22757;
        double r22759 = r22746 * r22736;
        double r22760 = r22758 + r22759;
        double r22761 = -r22732;
        double r22762 = r22761 - r22750;
        double r22763 = r22760 / r22762;
        double r22764 = -0.5;
        double r22765 = r22764 * r22737;
        double r22766 = r22755 ? r22763 : r22765;
        double r22767 = r22745 ? r22753 : r22766;
        double r22768 = r22734 ? r22743 : r22767;
        return r22768;
}

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 < -3.766148915908884e+131

    1. Initial program 55.0

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

    2. Taylor expanded around -inf 3.1

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

    if -3.766148915908884e+131 < b_2 < -2.9814431073825827e-294

    1. Initial program 9.4

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

    3. Applied clear-num9.5

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

    4. Simplified9.5

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

    if -2.9814431073825827e-294 < b_2 < 3.310436447834858e+94

    1. Initial program 31.1

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

    3. Applied div-inv31.1

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

    5. Applied flip-+31.1

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

    6. Applied associate-*l/31.2

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

    7. Simplified15.1

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

    9. Applied *-un-lft-identity15.1

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

    10. Applied *-un-lft-identity15.1

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

    11. Applied times-frac15.1

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

    12. Applied associate-*l*15.1

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

    13. Simplified8.8

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

    if 3.310436447834858e+94 < b_2

    1. Initial program 59.2

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

    2. Taylor expanded around inf 2.5

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.766148915908884171942137020871314382353 \cdot 10^{131}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -2.981443107382582658022271728267487907499 \cdot 10^{-294}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 3.310436447834858066850098837567764712113 \cdot 10^{94}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))