Average Error: 34.4 → 10.2
Time: 17.9s
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 r4265735 = b;
        double r4265736 = -r4265735;
        double r4265737 = r4265735 * r4265735;
        double r4265738 = 4.0;
        double r4265739 = a;
        double r4265740 = c;
        double r4265741 = r4265739 * r4265740;
        double r4265742 = r4265738 * r4265741;
        double r4265743 = r4265737 - r4265742;
        double r4265744 = sqrt(r4265743);
        double r4265745 = r4265736 + r4265744;
        double r4265746 = 2.0;
        double r4265747 = r4265746 * r4265739;
        double r4265748 = r4265745 / r4265747;
        return r4265748;
}
↓
double f(double a, double b, double c) {
        double r4265749 = b;
        double r4265750 = -1.7633154797394035e+89;
        bool r4265751 = r4265749 <= r4265750;
        double r4265752 = 2.0;
        double r4265753 = c;
        double r4265754 = r4265753 / r4265749;
        double r4265755 = r4265752 * r4265754;
        double r4265756 = a;
        double r4265757 = r4265749 / r4265756;
        double r4265758 = 2.0;
        double r4265759 = r4265757 * r4265758;
        double r4265760 = r4265755 - r4265759;
        double r4265761 = r4265760 / r4265752;
        double r4265762 = 9.136492990928292e-23;
        bool r4265763 = r4265749 <= r4265762;
        double r4265764 = r4265749 * r4265749;
        double r4265765 = 4.0;
        double r4265766 = r4265756 * r4265765;
        double r4265767 = r4265753 * r4265766;
        double r4265768 = r4265764 - r4265767;
        double r4265769 = sqrt(r4265768);
        double r4265770 = r4265769 - r4265749;
        double r4265771 = r4265770 / r4265756;
        double r4265772 = r4265771 / r4265752;
        double r4265773 = -2.0;
        double r4265774 = r4265773 * r4265754;
        double r4265775 = r4265774 / r4265752;
        double r4265776 = r4265763 ? r4265772 : r4265775;
        double r4265777 = r4265751 ? r4265761 : r4265776;
        return r4265777;
}

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)))