Average Error: 33.4 → 9.9
Time: 25.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.0027271082217074 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3864720 = b;
        double r3864721 = -r3864720;
        double r3864722 = r3864720 * r3864720;
        double r3864723 = 4.0;
        double r3864724 = a;
        double r3864725 = r3864723 * r3864724;
        double r3864726 = c;
        double r3864727 = r3864725 * r3864726;
        double r3864728 = r3864722 - r3864727;
        double r3864729 = sqrt(r3864728);
        double r3864730 = r3864721 + r3864729;
        double r3864731 = 2.0;
        double r3864732 = r3864731 * r3864724;
        double r3864733 = r3864730 / r3864732;
        return r3864733;
}
↓
double f(double a, double b, double c) {
        double r3864734 = b;
        double r3864735 = -1.0027271082217074e+110;
        bool r3864736 = r3864734 <= r3864735;
        double r3864737 = c;
        double r3864738 = r3864737 / r3864734;
        double r3864739 = a;
        double r3864740 = r3864734 / r3864739;
        double r3864741 = r3864738 - r3864740;
        double r3864742 = 2.0;
        double r3864743 = r3864741 * r3864742;
        double r3864744 = r3864743 / r3864742;
        double r3864745 = 2.326372645943808e-74;
        bool r3864746 = r3864734 <= r3864745;
        double r3864747 = -4.0;
        double r3864748 = r3864739 * r3864747;
        double r3864749 = r3864734 * r3864734;
        double r3864750 = fma(r3864737, r3864748, r3864749);
        double r3864751 = sqrt(r3864750);
        double r3864752 = r3864751 - r3864734;
        double r3864753 = r3864752 / r3864739;
        double r3864754 = r3864753 / r3864742;
        double r3864755 = -2.0;
        double r3864756 = r3864755 * r3864738;
        double r3864757 = r3864756 / r3864742;
        double r3864758 = r3864746 ? r3864754 : r3864757;
        double r3864759 = r3864736 ? r3864744 : r3864758;
        return r3864759;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.3
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0027271082217074e+110

    1. Initial program 46.7

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

    2. Simplified46.7

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -1.0027271082217074e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around inf 12.8

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

    4. Simplified12.8

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 8.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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)))