Average Error: 34.3 → 10.7
Time: 6.8s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.503298990987190001354942629806677715055 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 1.503298990987190001354942629806677715055 \cdot 10^{53}:\\
\;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r23001 = b_2;
        double r23002 = -r23001;
        double r23003 = r23001 * r23001;
        double r23004 = a;
        double r23005 = c;
        double r23006 = r23004 * r23005;
        double r23007 = r23003 - r23006;
        double r23008 = sqrt(r23007);
        double r23009 = r23002 + r23008;
        double r23010 = r23009 / r23004;
        return r23010;
}
↓
double f(double a, double b_2, double c) {
        double r23011 = b_2;
        double r23012 = -1.3696943711263392e-83;
        bool r23013 = r23011 <= r23012;
        double r23014 = 0.5;
        double r23015 = c;
        double r23016 = r23015 / r23011;
        double r23017 = r23014 * r23016;
        double r23018 = 2.0;
        double r23019 = a;
        double r23020 = r23011 / r23019;
        double r23021 = r23018 * r23020;
        double r23022 = r23017 - r23021;
        double r23023 = 1.50329899098719e+53;
        bool r23024 = r23011 <= r23023;
        double r23025 = 1.0;
        double r23026 = -r23011;
        double r23027 = r23011 * r23011;
        double r23028 = r23019 * r23015;
        double r23029 = r23027 - r23028;
        double r23030 = sqrt(r23029);
        double r23031 = r23026 - r23030;
        double r23032 = r23025 / r23031;
        double r23033 = cbrt(r23019);
        double r23034 = r23025 / r23015;
        double r23035 = r23033 / r23034;
        double r23036 = r23035 / r23033;
        double r23037 = r23032 * r23036;
        double r23038 = -0.5;
        double r23039 = r23038 * r23016;
        double r23040 = r23024 ? r23037 : r23039;
        double r23041 = r23013 ? r23022 : r23040;
        return r23041;
}

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 3 regimes

  2. if b_2 < -1.3696943711263392e-83

    1. Initial program 26.1

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

    2. Taylor expanded around -inf 12.3

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

    if -1.3696943711263392e-83 < b_2 < 1.50329899098719e+53

    1. Initial program 24.2

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

    3. Applied flip-+27.0

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

    4. Simplified17.7

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

    6. Applied *-un-lft-identity17.7

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

    7. Applied associate-/r*17.7

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

    8. Simplified16.0

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

    10. Applied add-cube-cbrt16.7

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

    11. Applied div-inv16.8

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

    12. Applied add-cube-cbrt16.1

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

    13. Applied times-frac16.3

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

    14. Applied times-frac14.1

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

    15. Simplified14.1

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

    if 1.50329899098719e+53 < b_2

    1. Initial program 57.4

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.503298990987190001354942629806677715055 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \frac{\frac{\sqrt[3]{a}}{\frac{1}{c}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))