Average Error: 34.2 → 9.4
Time: 13.0s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{\frac{1}{2} \cdot c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{-1}{2}}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;\left(\frac{\frac{1}{2} \cdot c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\
\;\;\;\;\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} - \frac{b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r19055 = b_2;
        double r19056 = -r19055;
        double r19057 = r19055 * r19055;
        double r19058 = a;
        double r19059 = c;
        double r19060 = r19058 * r19059;
        double r19061 = r19057 - r19060;
        double r19062 = sqrt(r19061);
        double r19063 = r19056 + r19062;
        double r19064 = r19063 / r19058;
        return r19064;
}
↓
double f(double a, double b_2, double c) {
        double r19065 = b_2;
        double r19066 = -3.7108875578650606e+138;
        bool r19067 = r19065 <= r19066;
        double r19068 = 0.5;
        double r19069 = c;
        double r19070 = r19068 * r19069;
        double r19071 = r19070 / r19065;
        double r19072 = a;
        double r19073 = r19065 / r19072;
        double r19074 = r19071 - r19073;
        double r19075 = r19074 - r19073;
        double r19076 = 4.626043257219638e-62;
        bool r19077 = r19065 <= r19076;
        double r19078 = 2.0;
        double r19079 = pow(r19065, r19078);
        double r19080 = r19072 * r19069;
        double r19081 = r19079 - r19080;
        double r19082 = sqrt(r19081);
        double r19083 = r19082 / r19072;
        double r19084 = r19083 - r19073;
        double r19085 = -0.5;
        double r19086 = r19069 * r19085;
        double r19087 = r19086 / r19065;
        double r19088 = r19077 ? r19084 : r19087;
        double r19089 = r19067 ? r19075 : r19088;
        return r19089;
}

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 < -3.7108875578650606e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    4. Applied div-sub58.5

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

    5. Simplified58.5

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

    6. Taylor expanded around -inf 2.0

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

    7. Simplified2.0

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

    if -3.7108875578650606e+138 < b_2 < 4.626043257219638e-62

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied div-sub12.1

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

    5. Simplified12.1

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

    if 4.626043257219638e-62 < b_2

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Taylor expanded around inf 8.4

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

    4. Simplified8.4

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{\frac{1}{2} \cdot c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{{b_2}^{2} - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{-1}{2}}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))