Average Error: 34.4 → 10.3
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, 2, \frac{b}{a} \cdot -2\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, 2, \frac{b}{a} \cdot -2\right)}{2}\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4718828 = b;
        double r4718829 = -r4718828;
        double r4718830 = r4718828 * r4718828;
        double r4718831 = 4.0;
        double r4718832 = a;
        double r4718833 = r4718831 * r4718832;
        double r4718834 = c;
        double r4718835 = r4718833 * r4718834;
        double r4718836 = r4718830 - r4718835;
        double r4718837 = sqrt(r4718836);
        double r4718838 = r4718829 + r4718837;
        double r4718839 = 2.0;
        double r4718840 = r4718839 * r4718832;
        double r4718841 = r4718838 / r4718840;
        return r4718841;
}
↓
double f(double a, double b, double c) {
        double r4718842 = b;
        double r4718843 = -1.7633154797394035e+89;
        bool r4718844 = r4718842 <= r4718843;
        double r4718845 = c;
        double r4718846 = r4718845 / r4718842;
        double r4718847 = 2.0;
        double r4718848 = a;
        double r4718849 = r4718842 / r4718848;
        double r4718850 = -2.0;
        double r4718851 = r4718849 * r4718850;
        double r4718852 = fma(r4718846, r4718847, r4718851);
        double r4718853 = r4718852 / r4718847;
        double r4718854 = 9.136492990928292e-23;
        bool r4718855 = r4718842 <= r4718854;
        double r4718856 = 1.0;
        double r4718857 = r4718856 / r4718848;
        double r4718858 = r4718842 * r4718842;
        double r4718859 = 4.0;
        double r4718860 = r4718859 * r4718845;
        double r4718861 = r4718848 * r4718860;
        double r4718862 = r4718858 - r4718861;
        double r4718863 = sqrt(r4718862);
        double r4718864 = r4718863 - r4718842;
        double r4718865 = r4718857 * r4718864;
        double r4718866 = r4718865 / r4718847;
        double r4718867 = -2.0;
        double r4718868 = r4718846 * r4718867;
        double r4718869 = r4718868 / r4718847;
        double r4718870 = r4718855 ? r4718866 : r4718869;
        double r4718871 = r4718844 ? r4718853 : r4718870;
        return r4718871;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    4. Applied div-inv45.8

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

    5. Taylor expanded around -inf 3.9

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

    6. Simplified3.9

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

    if -1.7633154797394035e+89 < b < 9.136492990928292e-23

    1. Initial program 15.0

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

    2. Simplified15.1

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

    4. Applied div-inv15.2

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.5

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

    2. Simplified55.4

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

    3. Taylor expanded around inf 6.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, 2, \frac{b}{a} \cdot -2\right)}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))