Average Error: 34.9 → 10.0
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.25165686884117225057308430661709452775 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -3.25165686884117225057308430661709452775 \cdot 10^{152}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4460882 = b;
        double r4460883 = -r4460882;
        double r4460884 = r4460882 * r4460882;
        double r4460885 = 4.0;
        double r4460886 = a;
        double r4460887 = c;
        double r4460888 = r4460886 * r4460887;
        double r4460889 = r4460885 * r4460888;
        double r4460890 = r4460884 - r4460889;
        double r4460891 = sqrt(r4460890);
        double r4460892 = r4460883 + r4460891;
        double r4460893 = 2.0;
        double r4460894 = r4460893 * r4460886;
        double r4460895 = r4460892 / r4460894;
        return r4460895;
}
↓
double f(double a, double b, double c) {
        double r4460896 = b;
        double r4460897 = -3.2516568688411723e+152;
        bool r4460898 = r4460896 <= r4460897;
        double r4460899 = c;
        double r4460900 = r4460899 / r4460896;
        double r4460901 = a;
        double r4460902 = r4460896 / r4460901;
        double r4460903 = r4460900 - r4460902;
        double r4460904 = 1.0;
        double r4460905 = r4460903 * r4460904;
        double r4460906 = 4.6129908231112306e-104;
        bool r4460907 = r4460896 <= r4460906;
        double r4460908 = r4460896 * r4460896;
        double r4460909 = 4.0;
        double r4460910 = r4460901 * r4460909;
        double r4460911 = r4460910 * r4460899;
        double r4460912 = r4460908 - r4460911;
        double r4460913 = sqrt(r4460912);
        double r4460914 = r4460913 - r4460896;
        double r4460915 = r4460914 / r4460901;
        double r4460916 = 2.0;
        double r4460917 = r4460915 / r4460916;
        double r4460918 = -1.0;
        double r4460919 = r4460900 * r4460918;
        double r4460920 = r4460907 ? r4460917 : r4460919;
        double r4460921 = r4460898 ? r4460905 : r4460920;
        return r4460921;
}

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

Original34.9
Target21.3
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.2516568688411723e+152

    1. Initial program 63.5

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

    2. Simplified63.5

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

    4. Applied *-un-lft-identity63.5

      \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{\color{blue}{1 \cdot 2}}\]

    5. Applied div-inv63.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \frac{1}{a}}}{1 \cdot 2}\]

    6. Applied times-frac63.5

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{1} \cdot \frac{\frac{1}{a}}{2}}\]

    7. Simplified63.5

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)} \cdot \frac{\frac{1}{a}}{2}\]
    8. Using strategy rm

    9. Applied associate-*r/63.5

      \[\leadsto \color{blue}{\frac{\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right) \cdot \frac{1}{a}}{2}}\]

    10. Simplified63.5

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

    11. Taylor expanded around -inf 2.3

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

    12. Simplified2.3

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

    if -3.2516568688411723e+152 < b < 4.6129908231112306e-104

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied *-un-lft-identity12.1

      \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{\color{blue}{1 \cdot 2}}\]

    5. Applied div-inv12.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b\right) \cdot \frac{1}{a}}}{1 \cdot 2}\]

    6. Applied times-frac12.3

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{1} \cdot \frac{\frac{1}{a}}{2}}\]

    7. Simplified12.2

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)} \cdot \frac{\frac{1}{a}}{2}\]
    8. Using strategy rm

    9. Applied associate-*r/12.2

      \[\leadsto \color{blue}{\frac{\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right) \cdot \frac{1}{a}}{2}}\]

    10. Simplified12.1

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

    if 4.6129908231112306e-104 < b

    1. Initial program 52.7

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

    2. Simplified52.7

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

    3. Taylor expanded around inf 9.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.25165686884117225057308430661709452775 \cdot 10^{152}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 4.612990823111230552052602417245542305295 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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)))