Average Error: 34.4 → 6.8
Time: 7.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.30101840923646093 \cdot 10^{98}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\
\;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r162722 = b;
        double r162723 = -r162722;
        double r162724 = r162722 * r162722;
        double r162725 = 4.0;
        double r162726 = a;
        double r162727 = c;
        double r162728 = r162726 * r162727;
        double r162729 = r162725 * r162728;
        double r162730 = r162724 - r162729;
        double r162731 = sqrt(r162730);
        double r162732 = r162723 + r162731;
        double r162733 = 2.0;
        double r162734 = r162733 * r162726;
        double r162735 = r162732 / r162734;
        return r162735;
}
↓
double f(double a, double b, double c) {
        double r162736 = b;
        double r162737 = -4.301018409236461e+98;
        bool r162738 = r162736 <= r162737;
        double r162739 = 1.0;
        double r162740 = c;
        double r162741 = r162740 / r162736;
        double r162742 = a;
        double r162743 = r162736 / r162742;
        double r162744 = r162741 - r162743;
        double r162745 = r162739 * r162744;
        double r162746 = -2.3757722518657493e-260;
        bool r162747 = r162736 <= r162746;
        double r162748 = -r162736;
        double r162749 = r162736 * r162736;
        double r162750 = 4.0;
        double r162751 = r162742 * r162740;
        double r162752 = r162750 * r162751;
        double r162753 = r162749 - r162752;
        double r162754 = sqrt(r162753);
        double r162755 = r162748 + r162754;
        double r162756 = 1.0;
        double r162757 = 2.0;
        double r162758 = r162757 * r162742;
        double r162759 = r162756 / r162758;
        double r162760 = r162755 * r162759;
        double r162761 = 6.6664567809045535e+68;
        bool r162762 = r162736 <= r162761;
        double r162763 = r162757 / r162750;
        double r162764 = r162756 / r162763;
        double r162765 = r162764 * r162740;
        double r162766 = r162748 - r162754;
        double r162767 = r162765 / r162766;
        double r162768 = -1.0;
        double r162769 = r162768 * r162741;
        double r162770 = r162762 ? r162767 : r162769;
        double r162771 = r162747 ? r162760 : r162770;
        double r162772 = r162738 ? r162745 : r162771;
        return r162772;
}

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
Target21.5
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.301018409236461e+98

    1. Initial program 47.2

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

    2. Taylor expanded around -inf 3.9

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

    3. Simplified3.9

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

    if -4.301018409236461e+98 < b < -2.3757722518657493e-260

    1. Initial program 8.5

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

    3. Applied div-inv8.7

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

    if -2.3757722518657493e-260 < b < 6.6664567809045535e+68

    1. Initial program 29.1

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

    3. Applied flip-+29.1

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

    4. Simplified16.2

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num16.4

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]

    7. Simplified15.9

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

    9. Applied associate-/r*15.8

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

    10. Simplified9.6

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

    12. Applied div-inv9.6

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

    13. Simplified9.5

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

    if 6.6664567809045535e+68 < b

    1. Initial program 58.7

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

    2. Taylor expanded around inf 3.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))