Average Error: 33.3 → 6.4
Time: 1.3m
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.15743459868086 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 4.901185863370092 \cdot 10^{+151}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.15743459868086 \cdot 10^{+131}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\

\mathbf{elif}\;b_2 \le -4.687918346756617 \cdot 10^{-254}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\

\mathbf{elif}\;b_2 \le 4.901185863370092 \cdot 10^{+151}:\\
\;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r4254740 = b_2;
        double r4254741 = -r4254740;
        double r4254742 = r4254740 * r4254740;
        double r4254743 = a;
        double r4254744 = c;
        double r4254745 = r4254743 * r4254744;
        double r4254746 = r4254742 - r4254745;
        double r4254747 = sqrt(r4254746);
        double r4254748 = r4254741 + r4254747;
        double r4254749 = r4254748 / r4254743;
        return r4254749;
}
↓
double f(double a, double b_2, double c) {
        double r4254750 = b_2;
        double r4254751 = -1.15743459868086e+131;
        bool r4254752 = r4254750 <= r4254751;
        double r4254753 = 0.5;
        double r4254754 = c;
        double r4254755 = r4254754 / r4254750;
        double r4254756 = r4254753 * r4254755;
        double r4254757 = a;
        double r4254758 = r4254750 / r4254757;
        double r4254759 = 2.0;
        double r4254760 = r4254758 * r4254759;
        double r4254761 = r4254756 - r4254760;
        double r4254762 = -4.687918346756617e-254;
        bool r4254763 = r4254750 <= r4254762;
        double r4254764 = 1.0;
        double r4254765 = r4254764 / r4254757;
        double r4254766 = r4254750 * r4254750;
        double r4254767 = r4254754 * r4254757;
        double r4254768 = r4254766 - r4254767;
        double r4254769 = sqrt(r4254768);
        double r4254770 = r4254769 - r4254750;
        double r4254771 = r4254765 * r4254770;
        double r4254772 = 4.901185863370092e+151;
        bool r4254773 = r4254750 <= r4254772;
        double r4254774 = -r4254754;
        double r4254775 = r4254769 + r4254750;
        double r4254776 = r4254774 / r4254775;
        double r4254777 = -0.5;
        double r4254778 = r4254777 * r4254755;
        double r4254779 = r4254773 ? r4254776 : r4254778;
        double r4254780 = r4254763 ? r4254771 : r4254779;
        double r4254781 = r4254752 ? r4254761 : r4254780;
        return r4254781;
}

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 < -1.15743459868086e+131

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around -inf 2.8

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

    if -1.15743459868086e+131 < b_2 < -4.687918346756617e-254

    1. Initial program 8.1

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

    2. Simplified8.1

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

    4. Applied div-inv8.2

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

    if -4.687918346756617e-254 < b_2 < 4.901185863370092e+151

    1. Initial program 32.8

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

    2. Simplified32.8

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

    4. Applied flip--32.9

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

    5. Simplified15.3

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

    7. Applied distribute-frac-neg15.3

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

    8. Applied distribute-frac-neg15.3

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

    9. Simplified8.4

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

    if 4.901185863370092e+151 < b_2

    1. Initial program 62.5

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

    2. Simplified62.5

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

    4. Applied div-inv62.5

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

    5. Taylor expanded around inf 1.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.15743459868086 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le -4.687918346756617 \cdot 10^{-254}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right)\\ \mathbf{elif}\;b_2 \le 4.901185863370092 \cdot 10^{+151}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))