Average Error: 34.7 → 10.7
Time: 38.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.900769547116861223219498082835437225018 \cdot 10^{46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 1.652881074072101299780548052808700933774 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -2.900769547116861223219498082835437225018 \cdot 10^{46}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, 2 \cdot \frac{c}{b}\right)}{2}\\

\mathbf{elif}\;b \le 1.652881074072101299780548052808700933774 \cdot 10^{-142}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3361860 = b;
        double r3361861 = -r3361860;
        double r3361862 = r3361860 * r3361860;
        double r3361863 = 4.0;
        double r3361864 = a;
        double r3361865 = c;
        double r3361866 = r3361864 * r3361865;
        double r3361867 = r3361863 * r3361866;
        double r3361868 = r3361862 - r3361867;
        double r3361869 = sqrt(r3361868);
        double r3361870 = r3361861 + r3361869;
        double r3361871 = 2.0;
        double r3361872 = r3361871 * r3361864;
        double r3361873 = r3361870 / r3361872;
        return r3361873;
}
↓
double f(double a, double b, double c) {
        double r3361874 = b;
        double r3361875 = -2.900769547116861e+46;
        bool r3361876 = r3361874 <= r3361875;
        double r3361877 = -2.0;
        double r3361878 = a;
        double r3361879 = r3361874 / r3361878;
        double r3361880 = 2.0;
        double r3361881 = c;
        double r3361882 = r3361881 / r3361874;
        double r3361883 = r3361880 * r3361882;
        double r3361884 = fma(r3361877, r3361879, r3361883);
        double r3361885 = r3361884 / r3361880;
        double r3361886 = 1.6528810740721013e-142;
        bool r3361887 = r3361874 <= r3361886;
        double r3361888 = 1.0;
        double r3361889 = r3361888 / r3361878;
        double r3361890 = r3361874 * r3361874;
        double r3361891 = 4.0;
        double r3361892 = r3361878 * r3361891;
        double r3361893 = r3361881 * r3361892;
        double r3361894 = r3361890 - r3361893;
        double r3361895 = sqrt(r3361894);
        double r3361896 = r3361895 - r3361874;
        double r3361897 = r3361889 * r3361896;
        double r3361898 = r3361897 / r3361880;
        double r3361899 = -2.0;
        double r3361900 = r3361882 * r3361899;
        double r3361901 = r3361900 / r3361880;
        double r3361902 = r3361887 ? r3361898 : r3361901;
        double r3361903 = r3361876 ? r3361885 : r3361902;
        return r3361903;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.7
Target21.1
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.900769547116861e+46

    1. Initial program 37.7

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

    2. Simplified37.7

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

    3. Taylor expanded around -inf 5.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified5.3

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

    if -2.900769547116861e+46 < b < 1.6528810740721013e-142

    1. Initial program 11.8

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

    2. Simplified11.7

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

    4. Applied div-inv11.9

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

    if 1.6528810740721013e-142 < b

    1. Initial program 50.7

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

    2. Simplified50.7

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

    3. Taylor expanded around inf 12.1

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.900769547116861223219498082835437225018 \cdot 10^{46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{b}{a}, 2 \cdot \frac{c}{b}\right)}{2}\\ \mathbf{elif}\;b \le 1.652881074072101299780548052808700933774 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))