Average Error: 33.7 → 10.0
Time: 19.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.8993775207591446 \cdot 10^{+126}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a} \cdot \frac{1}{2}\right) - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.6443485350724205 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -4.8993775207591446 \cdot 10^{+126}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a} \cdot \frac{1}{2}\right) - \frac{b}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.6443485350724205 \cdot 10^{-92}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2561734 = b;
        double r2561735 = -r2561734;
        double r2561736 = r2561734 * r2561734;
        double r2561737 = 4.0;
        double r2561738 = a;
        double r2561739 = c;
        double r2561740 = r2561738 * r2561739;
        double r2561741 = r2561737 * r2561740;
        double r2561742 = r2561736 - r2561741;
        double r2561743 = sqrt(r2561742);
        double r2561744 = r2561735 + r2561743;
        double r2561745 = 2.0;
        double r2561746 = r2561745 * r2561738;
        double r2561747 = r2561744 / r2561746;
        return r2561747;
}
↓
double f(double a, double b, double c) {
        double r2561748 = b;
        double r2561749 = -4.8993775207591446e+126;
        bool r2561750 = r2561748 <= r2561749;
        double r2561751 = c;
        double r2561752 = r2561751 / r2561748;
        double r2561753 = a;
        double r2561754 = r2561748 / r2561753;
        double r2561755 = 0.5;
        double r2561756 = r2561754 * r2561755;
        double r2561757 = r2561752 - r2561756;
        double r2561758 = 2.0;
        double r2561759 = r2561758 * r2561753;
        double r2561760 = r2561748 / r2561759;
        double r2561761 = r2561757 - r2561760;
        double r2561762 = 2.6443485350724205e-92;
        bool r2561763 = r2561748 <= r2561762;
        double r2561764 = r2561748 * r2561748;
        double r2561765 = 4.0;
        double r2561766 = r2561753 * r2561765;
        double r2561767 = r2561751 * r2561766;
        double r2561768 = r2561764 - r2561767;
        double r2561769 = sqrt(r2561768);
        double r2561770 = r2561769 / r2561759;
        double r2561771 = r2561770 - r2561760;
        double r2561772 = -r2561752;
        double r2561773 = r2561763 ? r2561771 : r2561772;
        double r2561774 = r2561750 ? r2561761 : r2561773;
        return r2561774;
}

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

Original33.7
Target20.5
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.8993775207591446e+126

    1. Initial program 51.7

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

    2. Simplified51.7

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

    4. Applied div-sub51.7

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

    5. Taylor expanded around -inf 3.5

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

    if -4.8993775207591446e+126 < b < 2.6443485350724205e-92

    1. Initial program 12.3

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

    2. Simplified12.2

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

    4. Applied div-sub12.2

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

    if 2.6443485350724205e-92 < b

    1. Initial program 51.9

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

    2. Simplified52.0

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

    4. Applied div-sub52.5

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

    5. Taylor expanded around inf 9.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    6. Simplified9.6

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.8993775207591446 \cdot 10^{+126}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a} \cdot \frac{1}{2}\right) - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.6443485350724205 \cdot 10^{-92}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))