Average Error: 32.9 → 22.1
Time: 2.7m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 2.399747870429879 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)} - b}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)}\right) \cdot a}\\ \end{array}\]
\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 2.399747870429879 \cdot 10^{-86}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)} - b}{a} \cdot \frac{1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)}\right) \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r7192825 = b;
        double r7192826 = -r7192825;
        double r7192827 = r7192825 * r7192825;
        double r7192828 = 4.0;
        double r7192829 = a;
        double r7192830 = r7192828 * r7192829;
        double r7192831 = c;
        double r7192832 = r7192830 * r7192831;
        double r7192833 = r7192827 - r7192832;
        double r7192834 = sqrt(r7192833);
        double r7192835 = r7192826 + r7192834;
        double r7192836 = 2.0;
        double r7192837 = r7192836 * r7192829;
        double r7192838 = r7192835 / r7192837;
        return r7192838;
}
↓
double f(double a, double b, double c) {
        double r7192839 = b;
        double r7192840 = 2.399747870429879e-86;
        bool r7192841 = r7192839 <= r7192840;
        double r7192842 = c;
        double r7192843 = -4.0;
        double r7192844 = r7192842 * r7192843;
        double r7192845 = a;
        double r7192846 = r7192839 * r7192839;
        double r7192847 = fma(r7192844, r7192845, r7192846);
        double r7192848 = sqrt(r7192847);
        double r7192849 = r7192848 - r7192839;
        double r7192850 = r7192849 / r7192845;
        double r7192851 = 0.5;
        double r7192852 = r7192850 * r7192851;
        double r7192853 = r7192844 * r7192845;
        double r7192854 = r7192839 + r7192848;
        double r7192855 = r7192854 * r7192845;
        double r7192856 = r7192853 / r7192855;
        double r7192857 = r7192851 * r7192856;
        double r7192858 = r7192841 ? r7192852 : r7192857;
        return r7192858;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < 2.399747870429879e-86

    1. Initial program 20.0

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

    2. Simplified20.0

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

    4. Applied *-un-lft-identity20.0

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

    5. Applied associate-/l*20.1

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

    7. Applied associate-/r/20.1

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

    8. Applied add-cube-cbrt20.1

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

    9. Applied times-frac20.1

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

    10. Simplified19.9

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

    11. Simplified19.9

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

    if 2.399747870429879e-86 < b

    1. Initial program 52.1

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

    2. Simplified52.2

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

    4. Applied *-un-lft-identity52.2

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

    5. Applied associate-/l*52.2

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

    7. Applied associate-/r/52.2

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

    8. Applied add-cube-cbrt52.2

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

    9. Applied times-frac52.2

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

    10. Simplified52.2

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

    11. Simplified52.2

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

    13. Applied flip--52.2

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

    14. Applied associate-/l/53.4

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

    15. Simplified25.2

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

  4. Final simplification22.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.399747870429879 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)} - b}{a} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(c \cdot -4\right) \cdot a}{\left(b + \sqrt{\mathsf{fma}\left(\left(c \cdot -4\right), a, \left(b \cdot b\right)\right)}\right) \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))