Average Error: 34.1 → 8.2
Time: 6.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\
\;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58846 = b;
        double r58847 = -r58846;
        double r58848 = r58846 * r58846;
        double r58849 = 4.0;
        double r58850 = a;
        double r58851 = r58849 * r58850;
        double r58852 = c;
        double r58853 = r58851 * r58852;
        double r58854 = r58848 - r58853;
        double r58855 = sqrt(r58854);
        double r58856 = r58847 + r58855;
        double r58857 = 2.0;
        double r58858 = r58857 * r58850;
        double r58859 = r58856 / r58858;
        return r58859;
}
↓
double f(double a, double b, double c) {
        double r58860 = b;
        double r58861 = -3.974595954042362e+78;
        bool r58862 = r58860 <= r58861;
        double r58863 = 1.0;
        double r58864 = c;
        double r58865 = r58864 / r58860;
        double r58866 = a;
        double r58867 = r58860 / r58866;
        double r58868 = r58865 - r58867;
        double r58869 = r58863 * r58868;
        double r58870 = -3.106669992168921e-249;
        bool r58871 = r58860 <= r58870;
        double r58872 = -r58860;
        double r58873 = r58860 * r58860;
        double r58874 = 4.0;
        double r58875 = r58874 * r58866;
        double r58876 = r58875 * r58864;
        double r58877 = r58873 - r58876;
        double r58878 = sqrt(r58877);
        double r58879 = r58872 + r58878;
        double r58880 = 1.0;
        double r58881 = 2.0;
        double r58882 = r58881 * r58866;
        double r58883 = r58880 / r58882;
        double r58884 = r58879 * r58883;
        double r58885 = 3.3294485045805704e-14;
        bool r58886 = r58860 <= r58885;
        double r58887 = r58880 / r58866;
        double r58888 = r58872 - r58878;
        double r58889 = r58888 / r58864;
        double r58890 = r58887 * r58889;
        double r58891 = r58874 / r58890;
        double r58892 = r58880 * r58891;
        double r58893 = r58892 / r58882;
        double r58894 = -1.0;
        double r58895 = r58894 * r58865;
        double r58896 = r58886 ? r58893 : r58895;
        double r58897 = r58871 ? r58884 : r58896;
        double r58898 = r58862 ? r58869 : r58897;
        return r58898;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -3.974595954042362e+78

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 4.2

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

    3. Simplified4.2

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

    if -3.974595954042362e+78 < b < -3.106669992168921e-249

    1. Initial program 8.2

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

    3. Applied div-inv8.3

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

    if -3.106669992168921e-249 < b < 3.3294485045805704e-14

    1. Initial program 23.6

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

    3. Applied flip-+23.7

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

    4. Simplified17.5

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

    6. Applied *-un-lft-identity17.5

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

    7. Applied *-un-lft-identity17.5

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

    8. Applied times-frac17.5

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

    9. Simplified17.5

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

    10. Simplified17.5

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

    12. Applied *-un-lft-identity17.5

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

    13. Applied times-frac14.8

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

    if 3.3294485045805704e-14 < b

    1. Initial program 55.3

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

    2. Taylor expanded around inf 5.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))