Average Error: 33.1 → 10.3
Time: 22.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1962309819144974 \cdot 10^{-65}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 5.6488521390017767 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -1.1962309819144974 \cdot 10^{-65}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 5.6488521390017767 \cdot 10^{+48}:\\
\;\;\;\;\frac{\frac{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2120778 = b;
        double r2120779 = -r2120778;
        double r2120780 = r2120778 * r2120778;
        double r2120781 = 4.0;
        double r2120782 = a;
        double r2120783 = c;
        double r2120784 = r2120782 * r2120783;
        double r2120785 = r2120781 * r2120784;
        double r2120786 = r2120780 - r2120785;
        double r2120787 = sqrt(r2120786);
        double r2120788 = r2120779 - r2120787;
        double r2120789 = 2.0;
        double r2120790 = r2120789 * r2120782;
        double r2120791 = r2120788 / r2120790;
        return r2120791;
}
↓
double f(double a, double b, double c) {
        double r2120792 = b;
        double r2120793 = -1.1962309819144974e-65;
        bool r2120794 = r2120792 <= r2120793;
        double r2120795 = -2.0;
        double r2120796 = c;
        double r2120797 = r2120796 / r2120792;
        double r2120798 = r2120795 * r2120797;
        double r2120799 = 2.0;
        double r2120800 = r2120798 / r2120799;
        double r2120801 = 5.6488521390017767e+48;
        bool r2120802 = r2120792 <= r2120801;
        double r2120803 = -4.0;
        double r2120804 = a;
        double r2120805 = r2120796 * r2120804;
        double r2120806 = r2120792 * r2120792;
        double r2120807 = fma(r2120803, r2120805, r2120806);
        double r2120808 = sqrt(r2120807);
        double r2120809 = -r2120808;
        double r2120810 = r2120809 - r2120792;
        double r2120811 = r2120810 / r2120804;
        double r2120812 = r2120811 / r2120799;
        double r2120813 = r2120792 / r2120804;
        double r2120814 = r2120797 - r2120813;
        double r2120815 = r2120814 * r2120799;
        double r2120816 = r2120815 / r2120799;
        double r2120817 = r2120802 ? r2120812 : r2120816;
        double r2120818 = r2120794 ? r2120800 : r2120817;
        return r2120818;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.1962309819144974e-65

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around -inf 8.8

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

    if -1.1962309819144974e-65 < b < 5.6488521390017767e+48

    1. Initial program 14.1

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

    2. Simplified14.1

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

    4. Applied *-un-lft-identity14.1

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

    5. Applied associate-/r*14.1

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

    6. Simplified14.1

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

    if 5.6488521390017767e+48 < b

    1. Initial program 35.6

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

    2. Simplified35.6

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

    3. Taylor expanded around inf 5.1

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

    4. Simplified5.1

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1962309819144974 \cdot 10^{-65}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 5.6488521390017767 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{\left(-\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}\right) - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))