Average Error: 34.4 → 10.6
Time: 20.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b}{a} \cdot -2\right)}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b}{a} \cdot -2\right)}{2}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3741828 = b;
        double r3741829 = -r3741828;
        double r3741830 = r3741828 * r3741828;
        double r3741831 = 4.0;
        double r3741832 = a;
        double r3741833 = r3741831 * r3741832;
        double r3741834 = c;
        double r3741835 = r3741833 * r3741834;
        double r3741836 = r3741830 - r3741835;
        double r3741837 = sqrt(r3741836);
        double r3741838 = r3741829 + r3741837;
        double r3741839 = 2.0;
        double r3741840 = r3741839 * r3741832;
        double r3741841 = r3741838 / r3741840;
        return r3741841;
}
↓
double f(double a, double b, double c) {
        double r3741842 = b;
        double r3741843 = -2.221067196710922e+149;
        bool r3741844 = r3741842 <= r3741843;
        double r3741845 = 2.0;
        double r3741846 = c;
        double r3741847 = r3741846 / r3741842;
        double r3741848 = a;
        double r3741849 = r3741842 / r3741848;
        double r3741850 = -2.0;
        double r3741851 = r3741849 * r3741850;
        double r3741852 = fma(r3741845, r3741847, r3741851);
        double r3741853 = r3741852 / r3741845;
        double r3741854 = 2.8983489306952693e-35;
        bool r3741855 = r3741842 <= r3741854;
        double r3741856 = r3741842 * r3741842;
        double r3741857 = 4.0;
        double r3741858 = r3741846 * r3741857;
        double r3741859 = r3741848 * r3741858;
        double r3741860 = r3741856 - r3741859;
        double r3741861 = sqrt(r3741860);
        double r3741862 = r3741861 - r3741842;
        double r3741863 = r3741862 / r3741848;
        double r3741864 = r3741863 / r3741845;
        double r3741865 = -2.0;
        double r3741866 = r3741865 * r3741847;
        double r3741867 = r3741866 / r3741845;
        double r3741868 = r3741855 ? r3741864 : r3741867;
        double r3741869 = r3741844 ? r3741853 : r3741868;
        return r3741869;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.5
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.8

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

    4. Simplified2.8

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

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.6

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

    2. Simplified14.6

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

    4. Applied *-un-lft-identity14.6

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

    5. Applied associate-/r*14.6

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

    6. Simplified14.6

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, \frac{c}{b}, \frac{b}{a} \cdot -2\right)}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))