Average Error: 34.9 → 8.2
Time: 18.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.806935659273273367110965907543014627108 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.421979387470590527625748567713748078502 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1813249.62920019752345979213714599609375:\\ \;\;\;\;\frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.806935659273273367110965907543014627108 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -1.421979387470590527625748567713748078502 \cdot 10^{-304}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{elif}\;b \le 1813249.62920019752345979213714599609375:\\
\;\;\;\;\frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r83900 = b;
        double r83901 = -r83900;
        double r83902 = r83900 * r83900;
        double r83903 = 4.0;
        double r83904 = a;
        double r83905 = c;
        double r83906 = r83904 * r83905;
        double r83907 = r83903 * r83906;
        double r83908 = r83902 - r83907;
        double r83909 = sqrt(r83908);
        double r83910 = r83901 + r83909;
        double r83911 = 2.0;
        double r83912 = r83911 * r83904;
        double r83913 = r83910 / r83912;
        return r83913;
}
↓
double f(double a, double b, double c) {
        double r83914 = b;
        double r83915 = -1.8069356592732734e+98;
        bool r83916 = r83914 <= r83915;
        double r83917 = 1.0;
        double r83918 = c;
        double r83919 = r83918 / r83914;
        double r83920 = a;
        double r83921 = r83914 / r83920;
        double r83922 = r83919 - r83921;
        double r83923 = r83917 * r83922;
        double r83924 = -1.4219793874705905e-304;
        bool r83925 = r83914 <= r83924;
        double r83926 = 1.0;
        double r83927 = 2.0;
        double r83928 = r83927 * r83920;
        double r83929 = r83914 * r83914;
        double r83930 = 4.0;
        double r83931 = r83920 * r83918;
        double r83932 = r83930 * r83931;
        double r83933 = r83929 - r83932;
        double r83934 = sqrt(r83933);
        double r83935 = r83934 - r83914;
        double r83936 = r83928 / r83935;
        double r83937 = r83926 / r83936;
        double r83938 = 1813249.6292001975;
        bool r83939 = r83914 <= r83938;
        double r83940 = r83930 * r83920;
        double r83941 = -r83914;
        double r83942 = r83941 - r83934;
        double r83943 = r83918 / r83942;
        double r83944 = r83940 * r83943;
        double r83945 = r83944 / r83928;
        double r83946 = -1.0;
        double r83947 = r83946 * r83919;
        double r83948 = r83939 ? r83945 : r83947;
        double r83949 = r83925 ? r83937 : r83948;
        double r83950 = r83916 ? r83923 : r83949;
        return r83950;
}

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
Herbie8.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8069356592732734e+98

    1. Initial program 46.8

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

    2. Taylor expanded around -inf 4.1

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

    3. Simplified4.1

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

    if -1.8069356592732734e+98 < b < -1.4219793874705905e-304

    1. Initial program 9.5

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

    3. Applied clear-num9.6

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

    4. Simplified9.6

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

    if -1.4219793874705905e-304 < b < 1813249.6292001975

    1. Initial program 27.1

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

    3. Applied flip-+27.2

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

    4. Simplified17.3

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity17.3

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

    7. Applied associate-/r*17.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{1}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    8. Simplified17.3

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

    10. Applied *-un-lft-identity17.3

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

    11. Applied times-frac14.3

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

    12. Simplified14.3

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

    if 1813249.6292001975 < b

    1. Initial program 56.8

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

    2. Taylor expanded around inf 4.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.806935659273273367110965907543014627108 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -1.421979387470590527625748567713748078502 \cdot 10^{-304}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1813249.62920019752345979213714599609375:\\ \;\;\;\;\frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))