Average Error: 34.2 → 6.8
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.39290314529454019 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.9276367402926466 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\\ \mathbf{elif}\;b \le 2.05720507804008149 \cdot 10^{80}:\\ \;\;\;\;\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{0.5} \cdot c\right) \cdot \frac{\sqrt[3]{1}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.39290314529454019 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -4.9276367402926466 \cdot 10^{-151}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\\

\mathbf{elif}\;b \le 2.05720507804008149 \cdot 10^{80}:\\
\;\;\;\;\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{0.5} \cdot c\right) \cdot \frac{\sqrt[3]{1}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r82006 = b;
        double r82007 = -r82006;
        double r82008 = r82006 * r82006;
        double r82009 = 4.0;
        double r82010 = a;
        double r82011 = c;
        double r82012 = r82010 * r82011;
        double r82013 = r82009 * r82012;
        double r82014 = r82008 - r82013;
        double r82015 = sqrt(r82014);
        double r82016 = r82007 + r82015;
        double r82017 = 2.0;
        double r82018 = r82017 * r82010;
        double r82019 = r82016 / r82018;
        return r82019;
}
↓
double f(double a, double b, double c) {
        double r82020 = b;
        double r82021 = -2.3929031452945402e+111;
        bool r82022 = r82020 <= r82021;
        double r82023 = 1.0;
        double r82024 = c;
        double r82025 = r82024 / r82020;
        double r82026 = a;
        double r82027 = r82020 / r82026;
        double r82028 = r82025 - r82027;
        double r82029 = r82023 * r82028;
        double r82030 = -4.9276367402926466e-151;
        bool r82031 = r82020 <= r82030;
        double r82032 = -r82020;
        double r82033 = r82020 * r82020;
        double r82034 = 4.0;
        double r82035 = r82026 * r82024;
        double r82036 = r82034 * r82035;
        double r82037 = r82033 - r82036;
        double r82038 = sqrt(r82037);
        double r82039 = r82032 + r82038;
        double r82040 = sqrt(r82039);
        double r82041 = 2.0;
        double r82042 = r82040 / r82041;
        double r82043 = r82040 / r82026;
        double r82044 = r82042 * r82043;
        double r82045 = 2.0572050780400815e+80;
        bool r82046 = r82020 <= r82045;
        double r82047 = 1.0;
        double r82048 = cbrt(r82047);
        double r82049 = r82048 * r82048;
        double r82050 = 0.5;
        double r82051 = r82049 / r82050;
        double r82052 = r82051 * r82024;
        double r82053 = r82032 - r82038;
        double r82054 = r82048 / r82053;
        double r82055 = r82052 * r82054;
        double r82056 = -1.0;
        double r82057 = r82056 * r82025;
        double r82058 = r82046 ? r82055 : r82057;
        double r82059 = r82031 ? r82044 : r82058;
        double r82060 = r82022 ? r82029 : r82059;
        return r82060;
}

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
Target20.8
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.3929031452945402e+111

    1. Initial program 49.3

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -2.3929031452945402e+111 < b < -4.9276367402926466e-151

    1. Initial program 6.4

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

    3. Applied add-sqr-sqrt6.8

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

    4. Applied times-frac6.8

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

    if -4.9276367402926466e-151 < b < 2.0572050780400815e+80

    1. Initial program 27.0

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

    3. Applied flip-+27.6

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

    4. Simplified16.3

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

    6. Applied *-un-lft-identity16.3

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

    7. Applied *-un-lft-identity16.3

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

    8. Applied times-frac16.3

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

    9. Applied associate-/l*16.4

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

    10. Simplified16.2

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

    11. Taylor expanded around 0 11.0

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{0.5}{c}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt11.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}\right) \cdot \sqrt[3]{\frac{1}{1}}}}{\frac{0.5}{c} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    14. Applied times-frac10.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}}{\frac{0.5}{c}} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    15. Simplified10.8

      \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{0.5} \cdot c\right)} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

    16. Simplified10.8

      \[\leadsto \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{0.5} \cdot c\right) \cdot \color{blue}{\frac{\sqrt[3]{1}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    if 2.0572050780400815e+80 < b

    1. Initial program 58.8

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

    2. Taylor expanded around inf 2.8

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.39290314529454019 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.9276367402926466 \cdot 10^{-151}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\\ \mathbf{elif}\;b \le 2.05720507804008149 \cdot 10^{80}:\\ \;\;\;\;\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{0.5} \cdot c\right) \cdot \frac{\sqrt[3]{1}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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)))