Average Error: 33.2 → 10.4
Time: 26.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.571206846913461 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c}{b} - \frac{b}{a}\right) - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -5.571206846913461 \cdot 10^{+106}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{c}{b} - \frac{b}{a}\right) - \frac{b}{a}}{2}\\

\mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2940861 = b;
        double r2940862 = -r2940861;
        double r2940863 = r2940861 * r2940861;
        double r2940864 = 4.0;
        double r2940865 = a;
        double r2940866 = r2940864 * r2940865;
        double r2940867 = c;
        double r2940868 = r2940866 * r2940867;
        double r2940869 = r2940863 - r2940868;
        double r2940870 = sqrt(r2940869);
        double r2940871 = r2940862 + r2940870;
        double r2940872 = 2.0;
        double r2940873 = r2940872 * r2940865;
        double r2940874 = r2940871 / r2940873;
        return r2940874;
}
↓
double f(double a, double b, double c) {
        double r2940875 = b;
        double r2940876 = -5.571206846913461e+106;
        bool r2940877 = r2940875 <= r2940876;
        double r2940878 = 2.0;
        double r2940879 = c;
        double r2940880 = r2940879 / r2940875;
        double r2940881 = r2940878 * r2940880;
        double r2940882 = a;
        double r2940883 = r2940875 / r2940882;
        double r2940884 = r2940881 - r2940883;
        double r2940885 = r2940884 - r2940883;
        double r2940886 = r2940885 / r2940878;
        double r2940887 = 3.821014310434392e-21;
        bool r2940888 = r2940875 <= r2940887;
        double r2940889 = -4.0;
        double r2940890 = r2940889 * r2940882;
        double r2940891 = r2940879 * r2940890;
        double r2940892 = fma(r2940875, r2940875, r2940891);
        double r2940893 = sqrt(r2940892);
        double r2940894 = r2940893 / r2940882;
        double r2940895 = r2940894 - r2940883;
        double r2940896 = r2940895 / r2940878;
        double r2940897 = -2.0;
        double r2940898 = r2940897 * r2940880;
        double r2940899 = r2940898 / r2940878;
        double r2940900 = r2940888 ? r2940896 : r2940899;
        double r2940901 = r2940877 ? r2940886 : r2940900;
        return r2940901;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.5
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.571206846913461e+106

    1. Initial program 46.5

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

    2. Simplified46.5

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

    4. Applied div-sub46.5

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

    5. Taylor expanded around -inf 3.5

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

    if -5.571206846913461e+106 < b < 3.821014310434392e-21

    1. Initial program 14.8

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

    2. Simplified14.8

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

    4. Applied div-sub14.8

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

    if 3.821014310434392e-21 < b

    1. Initial program 54.7

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

    2. Simplified54.7

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

    4. Applied div-sub55.4

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

    5. Taylor expanded around inf 6.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.571206846913461 \cdot 10^{+106}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{c}{b} - \frac{b}{a}\right) - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(-4 \cdot a\right)\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))