Average Error: 34.6 → 16.2
Time: 31.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\ \;\;\;\;\frac{\frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}} \cdot \left(\frac{a}{a} \cdot \frac{4 \cdot c}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-c\right), 4, 0\right)}{2 \cdot b}}{a}}{2}\\ \end{array}\]
\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 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}{a}}{2}\\

\mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\
\;\;\;\;\frac{\frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}} \cdot \left(\frac{a}{a} \cdot \frac{4 \cdot c}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-c\right), 4, 0\right)}{2 \cdot b}}{a}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r51897 = b;
        double r51898 = -r51897;
        double r51899 = r51897 * r51897;
        double r51900 = 4.0;
        double r51901 = a;
        double r51902 = r51900 * r51901;
        double r51903 = c;
        double r51904 = r51902 * r51903;
        double r51905 = r51899 - r51904;
        double r51906 = sqrt(r51905);
        double r51907 = r51898 + r51906;
        double r51908 = 2.0;
        double r51909 = r51908 * r51901;
        double r51910 = r51907 / r51909;
        return r51910;
}
↓
double f(double a, double b, double c) {
        double r51911 = b;
        double r51912 = 2.4944562401296086e-289;
        bool r51913 = r51911 <= r51912;
        double r51914 = a;
        double r51915 = -r51914;
        double r51916 = 4.0;
        double r51917 = r51915 * r51916;
        double r51918 = c;
        double r51919 = r51911 * r51911;
        double r51920 = fma(r51917, r51918, r51919);
        double r51921 = sqrt(r51920);
        double r51922 = r51921 - r51911;
        double r51923 = r51922 / r51914;
        double r51924 = 2.0;
        double r51925 = r51923 / r51924;
        double r51926 = 3.224491050532555e+112;
        bool r51927 = r51911 <= r51926;
        double r51928 = -1.0;
        double r51929 = -r51918;
        double r51930 = r51914 * r51929;
        double r51931 = fma(r51916, r51930, r51919);
        double r51932 = sqrt(r51931);
        double r51933 = r51932 + r51911;
        double r51934 = cbrt(r51933);
        double r51935 = r51928 / r51934;
        double r51936 = r51935 / r51934;
        double r51937 = r51914 / r51914;
        double r51938 = r51916 * r51918;
        double r51939 = r51938 / r51934;
        double r51940 = r51937 * r51939;
        double r51941 = r51936 * r51940;
        double r51942 = r51941 / r51924;
        double r51943 = 0.0;
        double r51944 = fma(r51930, r51916, r51943);
        double r51945 = 2.0;
        double r51946 = r51945 * r51911;
        double r51947 = r51944 / r51946;
        double r51948 = r51947 / r51914;
        double r51949 = r51948 / r51924;
        double r51950 = r51927 ? r51942 : r51949;
        double r51951 = r51913 ? r51925 : r51950;
        return r51951;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < 2.4944562401296086e-289

    1. Initial program 22.5

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

    2. Simplified22.5

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

    4. Applied *-un-lft-identity22.5

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

    5. Applied *-un-lft-identity22.5

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

    6. Applied times-frac22.5

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

    7. Simplified22.5

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

    8. Simplified22.5

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

    if 2.4944562401296086e-289 < b < 3.224491050532555e+112

    1. Initial program 33.8

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

    2. Simplified33.8

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

    4. Applied flip--33.8

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

    5. Simplified15.8

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

    6. Simplified15.8

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

    8. Applied *-un-lft-identity15.8

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

    9. Applied add-cube-cbrt16.5

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}}{1 \cdot a}}{2}\]

    10. Applied *-un-lft-identity16.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

    11. Applied times-frac16.5

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}} \cdot \frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}}{1 \cdot a}}{2}\]

    12. Applied times-frac15.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{1} \cdot \frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{a}}}{2}\]

    13. Simplified15.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}} \cdot \frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{a}}{2}\]

    14. Simplified9.1

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

    if 3.224491050532555e+112 < b

    1. Initial program 60.8

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

    2. Simplified60.8

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

    4. Applied flip--60.8

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

    5. Simplified32.5

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

    6. Simplified32.5

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

    7. Taylor expanded around 0 13.1

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{2 \cdot b}}}{a}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\ \;\;\;\;\frac{\frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}} \cdot \left(\frac{a}{a} \cdot \frac{4 \cdot c}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-c\right), 4, 0\right)}{2 \cdot b}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))