Average Error: 34.2 → 10.4
Time: 13.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.12310353364421125 \cdot 10^{95}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\
\;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r89985 = b;
        double r89986 = -r89985;
        double r89987 = r89985 * r89985;
        double r89988 = 4.0;
        double r89989 = a;
        double r89990 = c;
        double r89991 = r89989 * r89990;
        double r89992 = r89988 * r89991;
        double r89993 = r89987 - r89992;
        double r89994 = sqrt(r89993);
        double r89995 = r89986 + r89994;
        double r89996 = 2.0;
        double r89997 = r89996 * r89989;
        double r89998 = r89995 / r89997;
        return r89998;
}
↓
double f(double a, double b, double c) {
        double r89999 = b;
        double r90000 = -4.123103533644211e+95;
        bool r90001 = r89999 <= r90000;
        double r90002 = 1.0;
        double r90003 = c;
        double r90004 = r90003 / r89999;
        double r90005 = a;
        double r90006 = r89999 / r90005;
        double r90007 = r90004 - r90006;
        double r90008 = r90002 * r90007;
        double r90009 = 3.446447862996811e-75;
        bool r90010 = r89999 <= r90009;
        double r90011 = r89999 * r89999;
        double r90012 = 4.0;
        double r90013 = r90005 * r90003;
        double r90014 = r90012 * r90013;
        double r90015 = r90011 - r90014;
        double r90016 = sqrt(r90015);
        double r90017 = r90016 - r89999;
        double r90018 = 1.0;
        double r90019 = 2.0;
        double r90020 = r90018 / r90019;
        double r90021 = r90005 / r90020;
        double r90022 = r90018 / r90021;
        double r90023 = r90017 * r90022;
        double r90024 = -1.0;
        double r90025 = r90024 * r90004;
        double r90026 = r90010 ? r90023 : r90025;
        double r90027 = r90001 ? r90008 : r90026;
        return r90027;
}

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
Target21.1
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.123103533644211e+95

    1. Initial program 47.3

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

    2. Simplified47.3

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

    3. Taylor expanded around -inf 3.8

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

    4. Simplified3.8

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

    if -4.123103533644211e+95 < b < 3.446447862996811e-75

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied *-un-lft-identity13.3

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

    5. Applied *-un-lft-identity13.3

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

    6. Applied times-frac13.3

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

    7. Applied associate-/l*13.4

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

    9. Applied div-inv13.4

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

    10. Applied *-un-lft-identity13.4

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

    11. Applied times-frac13.5

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

    12. Applied add-cube-cbrt13.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}\right) \cdot \sqrt[3]{\frac{1}{1}}}}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{a}{\frac{1}{2}}}\]

    13. Applied times-frac13.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\frac{a}{\frac{1}{2}}}}\]

    14. Simplified13.4

      \[\leadsto \color{blue}{\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\frac{a}{\frac{1}{2}}}\]

    15. Simplified13.4

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

    if 3.446447862996811e-75 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 9.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.12310353364421125 \cdot 10^{95}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.446447862996811 \cdot 10^{-75}:\\ \;\;\;\;\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right) \cdot \frac{1}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))