Average Error: 32.9 → 22.0
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.0362819647114397 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{a}}{2}\\ \end{array}\]
\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 1.0362819647114397 \cdot 10^{-153}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1848839 = b;
        double r1848840 = -r1848839;
        double r1848841 = r1848839 * r1848839;
        double r1848842 = 4.0;
        double r1848843 = a;
        double r1848844 = c;
        double r1848845 = r1848843 * r1848844;
        double r1848846 = r1848842 * r1848845;
        double r1848847 = r1848841 - r1848846;
        double r1848848 = sqrt(r1848847);
        double r1848849 = r1848840 + r1848848;
        double r1848850 = 2.0;
        double r1848851 = r1848850 * r1848843;
        double r1848852 = r1848849 / r1848851;
        return r1848852;
}
↓
double f(double a, double b, double c) {
        double r1848853 = b;
        double r1848854 = 1.0362819647114397e-153;
        bool r1848855 = r1848853 <= r1848854;
        double r1848856 = a;
        double r1848857 = c;
        double r1848858 = r1848856 * r1848857;
        double r1848859 = -4.0;
        double r1848860 = r1848853 * r1848853;
        double r1848861 = fma(r1848858, r1848859, r1848860);
        double r1848862 = sqrt(r1848861);
        double r1848863 = sqrt(r1848862);
        double r1848864 = -r1848853;
        double r1848865 = fma(r1848863, r1848863, r1848864);
        double r1848866 = r1848865 / r1848856;
        double r1848867 = 2.0;
        double r1848868 = r1848866 / r1848867;
        double r1848869 = 0.0;
        double r1848870 = fma(r1848858, r1848859, r1848869);
        double r1848871 = r1848853 + r1848862;
        double r1848872 = r1848870 / r1848871;
        double r1848873 = r1848872 / r1848856;
        double r1848874 = r1848873 / r1848867;
        double r1848875 = r1848855 ? r1848868 : r1848874;
        return r1848875;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.4
Herbie22.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 1.0362819647114397e-153

    1. Initial program 19.8

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

    2. Simplified19.8

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

    4. Applied add-sqr-sqrt19.8

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

    5. Applied sqrt-prod20.0

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

    6. Applied fma-neg19.9

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

    if 1.0362819647114397e-153 < b

    1. Initial program 49.2

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

    2. Simplified49.1

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

    4. Applied flip--49.2

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

    5. Simplified24.6

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

  4. Final simplification22.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.0362819647114397 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{b + \sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))