Average Error: 33.6 → 10.4
Time: 15.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6262848 = b;
        double r6262849 = -r6262848;
        double r6262850 = r6262848 * r6262848;
        double r6262851 = 4.0;
        double r6262852 = a;
        double r6262853 = r6262851 * r6262852;
        double r6262854 = c;
        double r6262855 = r6262853 * r6262854;
        double r6262856 = r6262850 - r6262855;
        double r6262857 = sqrt(r6262856);
        double r6262858 = r6262849 + r6262857;
        double r6262859 = 2.0;
        double r6262860 = r6262859 * r6262852;
        double r6262861 = r6262858 / r6262860;
        return r6262861;
}
↓
double f(double a, double b, double c) {
        double r6262862 = b;
        double r6262863 = -2.1144981103869975e+131;
        bool r6262864 = r6262862 <= r6262863;
        double r6262865 = c;
        double r6262866 = r6262865 / r6262862;
        double r6262867 = a;
        double r6262868 = r6262862 / r6262867;
        double r6262869 = r6262866 - r6262868;
        double r6262870 = 4.5810084990875205e-68;
        bool r6262871 = r6262862 <= r6262870;
        double r6262872 = 1.0;
        double r6262873 = 2.0;
        double r6262874 = r6262867 * r6262873;
        double r6262875 = -4.0;
        double r6262876 = r6262865 * r6262875;
        double r6262877 = r6262876 * r6262867;
        double r6262878 = fma(r6262862, r6262862, r6262877);
        double r6262879 = sqrt(r6262878);
        double r6262880 = r6262879 - r6262862;
        double r6262881 = r6262874 / r6262880;
        double r6262882 = r6262872 / r6262881;
        double r6262883 = -r6262866;
        double r6262884 = r6262871 ? r6262882 : r6262883;
        double r6262885 = r6262864 ? r6262869 : r6262884;
        return r6262885;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target21.0
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 < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    3. Applied clear-num13.4

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

    4. Simplified13.4

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

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -4\right) \cdot a\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))