Average Error: 33.6 → 10.4
Time: 16.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2993666 = b;
        double r2993667 = -r2993666;
        double r2993668 = r2993666 * r2993666;
        double r2993669 = 4.0;
        double r2993670 = a;
        double r2993671 = c;
        double r2993672 = r2993670 * r2993671;
        double r2993673 = r2993669 * r2993672;
        double r2993674 = r2993668 - r2993673;
        double r2993675 = sqrt(r2993674);
        double r2993676 = r2993667 + r2993675;
        double r2993677 = 2.0;
        double r2993678 = r2993677 * r2993670;
        double r2993679 = r2993676 / r2993678;
        return r2993679;
}
↓
double f(double a, double b, double c) {
        double r2993680 = b;
        double r2993681 = -2.1144981103869975e+131;
        bool r2993682 = r2993680 <= r2993681;
        double r2993683 = c;
        double r2993684 = r2993683 / r2993680;
        double r2993685 = a;
        double r2993686 = r2993680 / r2993685;
        double r2993687 = r2993684 - r2993686;
        double r2993688 = 2.0;
        double r2993689 = r2993687 * r2993688;
        double r2993690 = r2993689 / r2993688;
        double r2993691 = 4.5810084990875205e-68;
        bool r2993692 = r2993680 <= r2993691;
        double r2993693 = 1.0;
        double r2993694 = r2993680 * r2993680;
        double r2993695 = 4.0;
        double r2993696 = r2993695 * r2993685;
        double r2993697 = r2993696 * r2993683;
        double r2993698 = r2993694 - r2993697;
        double r2993699 = sqrt(r2993698);
        double r2993700 = r2993699 - r2993680;
        double r2993701 = r2993685 / r2993700;
        double r2993702 = r2993693 / r2993701;
        double r2993703 = r2993702 / r2993688;
        double r2993704 = -2.0;
        double r2993705 = r2993704 * r2993684;
        double r2993706 = r2993705 / r2993688;
        double r2993707 = r2993692 ? r2993703 : r2993706;
        double r2993708 = r2993682 ? r2993690 : r2993707;
        return r2993708;
}

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

Original33.6
Target21.0
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Simplified53.8

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

    3. Taylor expanded around -inf 2.6

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

    4. Simplified2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    2. Simplified13.3

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

    4. Applied clear-num13.5

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

    if 4.5810084990875205e-68 < 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

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))