Average Error: 34.2 → 6.6
Time: 4.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.4767676629755074 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.4767676629755074 \cdot 10^{150}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r84925 = b;
        double r84926 = -r84925;
        double r84927 = r84925 * r84925;
        double r84928 = 4.0;
        double r84929 = a;
        double r84930 = c;
        double r84931 = r84929 * r84930;
        double r84932 = r84928 * r84931;
        double r84933 = r84927 - r84932;
        double r84934 = sqrt(r84933);
        double r84935 = r84926 - r84934;
        double r84936 = 2.0;
        double r84937 = r84936 * r84929;
        double r84938 = r84935 / r84937;
        return r84938;
}
↓
double f(double a, double b, double c) {
        double r84939 = b;
        double r84940 = -4.4767676629755074e+150;
        bool r84941 = r84939 <= r84940;
        double r84942 = -1.0;
        double r84943 = c;
        double r84944 = r84943 / r84939;
        double r84945 = r84942 * r84944;
        double r84946 = -9.7864508816149e-248;
        bool r84947 = r84939 <= r84946;
        double r84948 = 1.0;
        double r84949 = 2.0;
        double r84950 = r84948 / r84949;
        double r84951 = 4.0;
        double r84952 = r84951 * r84943;
        double r84953 = r84939 * r84939;
        double r84954 = a;
        double r84955 = r84954 * r84943;
        double r84956 = r84951 * r84955;
        double r84957 = r84953 - r84956;
        double r84958 = sqrt(r84957);
        double r84959 = r84958 - r84939;
        double r84960 = r84952 / r84959;
        double r84961 = r84950 * r84960;
        double r84962 = 8.293479107059485e+98;
        bool r84963 = r84939 <= r84962;
        double r84964 = -r84939;
        double r84965 = r84949 * r84954;
        double r84966 = r84964 / r84965;
        double r84967 = r84958 / r84965;
        double r84968 = r84966 - r84967;
        double r84969 = 1.0;
        double r84970 = r84939 / r84954;
        double r84971 = r84944 - r84970;
        double r84972 = r84969 * r84971;
        double r84973 = r84963 ? r84968 : r84972;
        double r84974 = r84947 ? r84961 : r84973;
        double r84975 = r84941 ? r84945 : r84974;
        return r84975;
}

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
Target21.5
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.4767676629755074e+150

    1. Initial program 63.7

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

    2. Taylor expanded around -inf 1.4

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

    if -4.4767676629755074e+150 < b < -9.7864508816149e-248

    1. Initial program 36.5

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

    3. Applied flip--36.5

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

    4. Simplified16.2

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

    5. Simplified16.2

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

    7. Applied *-un-lft-identity16.2

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

    8. Applied *-un-lft-identity16.2

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

    9. Applied times-frac16.2

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

    10. Applied times-frac16.2

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

    11. Simplified16.2

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

    12. Simplified14.2

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

    14. Applied *-un-lft-identity14.2

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

    15. Applied times-frac14.2

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

    16. Simplified14.2

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

    17. Simplified7.3

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

    if -9.7864508816149e-248 < b < 8.293479107059485e+98

    1. Initial program 10.0

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

    3. Applied div-sub10.0

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

    if 8.293479107059485e+98 < b

    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 3.8

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

    3. Simplified3.8

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.4767676629755074 \cdot 10^{150}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.7864508816148999 \cdot 10^{-248}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 8.29347910705948456 \cdot 10^{98}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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