Average Error: 34.2 → 9.5
Time: 15.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a} - \frac{\frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a} - \frac{\frac{b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r5572887 = b;
        double r5572888 = -r5572887;
        double r5572889 = r5572887 * r5572887;
        double r5572890 = 4.0;
        double r5572891 = a;
        double r5572892 = c;
        double r5572893 = r5572891 * r5572892;
        double r5572894 = r5572890 * r5572893;
        double r5572895 = r5572889 - r5572894;
        double r5572896 = sqrt(r5572895);
        double r5572897 = r5572888 + r5572896;
        double r5572898 = 2.0;
        double r5572899 = r5572898 * r5572891;
        double r5572900 = r5572897 / r5572899;
        return r5572900;
}
↓
double f(double a, double b, double c) {
        double r5572901 = b;
        double r5572902 = -3.7108875578650606e+138;
        bool r5572903 = r5572901 <= r5572902;
        double r5572904 = 1.0;
        double r5572905 = c;
        double r5572906 = r5572905 / r5572901;
        double r5572907 = a;
        double r5572908 = r5572901 / r5572907;
        double r5572909 = r5572906 - r5572908;
        double r5572910 = r5572904 * r5572909;
        double r5572911 = 4.626043257219638e-62;
        bool r5572912 = r5572901 <= r5572911;
        double r5572913 = r5572901 * r5572901;
        double r5572914 = 4.0;
        double r5572915 = r5572907 * r5572905;
        double r5572916 = r5572914 * r5572915;
        double r5572917 = r5572913 - r5572916;
        double r5572918 = sqrt(r5572917);
        double r5572919 = 2.0;
        double r5572920 = r5572918 / r5572919;
        double r5572921 = r5572920 / r5572907;
        double r5572922 = r5572901 / r5572919;
        double r5572923 = r5572922 / r5572907;
        double r5572924 = r5572921 - r5572923;
        double r5572925 = -1.0;
        double r5572926 = r5572925 * r5572906;
        double r5572927 = r5572912 ? r5572924 : r5572926;
        double r5572928 = r5572903 ? r5572910 : r5572927;
        return r5572928;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.2
Target21.0
Herbie9.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.7108875578650606e+138

    1. Initial program 58.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified58.5

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}}\]

    3. Taylor expanded around -inf 2.0

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

    4. Simplified2.0

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

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

    1. Initial program 12.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified12.3

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}}\]
    3. Using strategy rm

    4. Applied div-sub12.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2} - \frac{b}{2}}}{a}\]

    5. Applied div-sub12.3

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a} - \frac{\frac{b}{2}}{a}}\]

    if 4.626043257219638e-62 < b

    1. Initial program 53.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Simplified53.7

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}}\]

    3. Taylor expanded around inf 8.5

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.626043257219637986942022736183111936335 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a} - \frac{\frac{b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))