Average Error: 34.4 → 10.2
Time: 18.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.763315479739403460017265344144602342789 \cdot 10^{89}:\\
\;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\

\mathbf{elif}\;b \le 9.136492990928292133394320076175633285536 \cdot 10^{-23}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \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 r4509700 = b;
        double r4509701 = -r4509700;
        double r4509702 = r4509700 * r4509700;
        double r4509703 = 4.0;
        double r4509704 = a;
        double r4509705 = c;
        double r4509706 = r4509704 * r4509705;
        double r4509707 = r4509703 * r4509706;
        double r4509708 = r4509702 - r4509707;
        double r4509709 = sqrt(r4509708);
        double r4509710 = r4509701 + r4509709;
        double r4509711 = 2.0;
        double r4509712 = r4509711 * r4509704;
        double r4509713 = r4509710 / r4509712;
        return r4509713;
}
↓
double f(double a, double b, double c) {
        double r4509714 = b;
        double r4509715 = -1.7633154797394035e+89;
        bool r4509716 = r4509714 <= r4509715;
        double r4509717 = 2.0;
        double r4509718 = c;
        double r4509719 = r4509718 / r4509714;
        double r4509720 = r4509717 * r4509719;
        double r4509721 = a;
        double r4509722 = r4509714 / r4509721;
        double r4509723 = 2.0;
        double r4509724 = r4509722 * r4509723;
        double r4509725 = r4509720 - r4509724;
        double r4509726 = r4509725 / r4509717;
        double r4509727 = 9.136492990928292e-23;
        bool r4509728 = r4509714 <= r4509727;
        double r4509729 = r4509714 * r4509714;
        double r4509730 = 4.0;
        double r4509731 = r4509721 * r4509730;
        double r4509732 = r4509718 * r4509731;
        double r4509733 = r4509729 - r4509732;
        double r4509734 = sqrt(r4509733);
        double r4509735 = r4509734 - r4509714;
        double r4509736 = r4509735 / r4509721;
        double r4509737 = r4509736 / r4509717;
        double r4509738 = -2.0;
        double r4509739 = r4509738 * r4509719;
        double r4509740 = r4509739 / r4509717;
        double r4509741 = r4509728 ? r4509737 : r4509740;
        double r4509742 = r4509716 ? r4509726 : r4509741;
        return r4509742;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if b < -1.7633154797394035e+89

    1. Initial program 45.7

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

    2. Simplified45.7

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

    3. Taylor expanded around -inf 3.9

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

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

    1. Initial program 15.0

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

    2. Simplified15.0

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

    4. Applied div-inv15.2

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

    6. Applied un-div-inv15.0

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

    if 9.136492990928292e-23 < b

    1. Initial program 55.4

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

    2. Simplified55.5

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

    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 simplification10.2

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

Reproduce

herbie shell --seed 2019172 
(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)))