Average Error: 34.1 → 8.7
Time: 8.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.8371925747446876 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.14106444929204386 \cdot 10^{-250}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.78898484253553231 \cdot 10^{48}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -4.8371925747446876 \cdot 10^{53}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -8.14106444929204386 \cdot 10^{-250}:\\
\;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\

\mathbf{elif}\;b_2 \le 1.78898484253553231 \cdot 10^{48}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r34039 = b_2;
        double r34040 = -r34039;
        double r34041 = r34039 * r34039;
        double r34042 = a;
        double r34043 = c;
        double r34044 = r34042 * r34043;
        double r34045 = r34041 - r34044;
        double r34046 = sqrt(r34045);
        double r34047 = r34040 + r34046;
        double r34048 = r34047 / r34042;
        return r34048;
}
↓
double f(double a, double b_2, double c) {
        double r34049 = b_2;
        double r34050 = -4.837192574744688e+53;
        bool r34051 = r34049 <= r34050;
        double r34052 = 0.5;
        double r34053 = c;
        double r34054 = r34053 / r34049;
        double r34055 = r34052 * r34054;
        double r34056 = 2.0;
        double r34057 = a;
        double r34058 = r34049 / r34057;
        double r34059 = r34056 * r34058;
        double r34060 = r34055 - r34059;
        double r34061 = -8.141064449292044e-250;
        bool r34062 = r34049 <= r34061;
        double r34063 = 1.0;
        double r34064 = r34049 * r34049;
        double r34065 = r34057 * r34053;
        double r34066 = r34064 - r34065;
        double r34067 = sqrt(r34066);
        double r34068 = r34067 - r34049;
        double r34069 = r34057 / r34068;
        double r34070 = r34063 / r34069;
        double r34071 = 1.7889848425355323e+48;
        bool r34072 = r34049 <= r34071;
        double r34073 = -r34049;
        double r34074 = r34073 - r34067;
        double r34075 = r34074 / r34057;
        double r34076 = r34075 / r34053;
        double r34077 = r34063 / r34076;
        double r34078 = r34077 / r34057;
        double r34079 = -0.5;
        double r34080 = r34079 * r34054;
        double r34081 = r34072 ? r34078 : r34080;
        double r34082 = r34062 ? r34070 : r34081;
        double r34083 = r34051 ? r34060 : r34082;
        return r34083;
}

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 < -4.837192574744688e+53

    1. Initial program 37.6

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

    2. Taylor expanded around -inf 5.6

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

    if -4.837192574744688e+53 < b_2 < -8.141064449292044e-250

    1. Initial program 9.3

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

    3. Applied clear-num9.4

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

    4. Simplified9.4

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

    if -8.141064449292044e-250 < b_2 < 1.7889848425355323e+48

    1. Initial program 27.4

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

    3. Applied flip-+27.4

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

    4. Simplified16.4

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

    6. Applied clear-num16.5

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

    7. Simplified14.5

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

    if 1.7889848425355323e+48 < b_2

    1. Initial program 57.6

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

    2. Taylor expanded around inf 4.4

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.8371925747446876 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.14106444929204386 \cdot 10^{-250}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 1.78898484253553231 \cdot 10^{48}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))