Average Error: 33.9 → 8.8
Time: 7.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.896807826547109987464579044865965135876 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 1.505965056562671542641677182626911662098 \cdot 10^{74}:\\ \;\;\;\;\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.896807826547109987464579044865965135876 \cdot 10^{-222}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{elif}\;b \le 1.505965056562671542641677182626911662098 \cdot 10^{74}:\\
\;\;\;\;\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r61643 = b;
        double r61644 = -r61643;
        double r61645 = r61643 * r61643;
        double r61646 = 4.0;
        double r61647 = a;
        double r61648 = r61646 * r61647;
        double r61649 = c;
        double r61650 = r61648 * r61649;
        double r61651 = r61645 - r61650;
        double r61652 = sqrt(r61651);
        double r61653 = r61644 + r61652;
        double r61654 = 2.0;
        double r61655 = r61654 * r61647;
        double r61656 = r61653 / r61655;
        return r61656;
}
↓
double f(double a, double b, double c) {
        double r61657 = b;
        double r61658 = -1.361733299857302e+105;
        bool r61659 = r61657 <= r61658;
        double r61660 = 1.0;
        double r61661 = c;
        double r61662 = r61661 / r61657;
        double r61663 = a;
        double r61664 = r61657 / r61663;
        double r61665 = r61662 - r61664;
        double r61666 = r61660 * r61665;
        double r61667 = 6.89680782654711e-222;
        bool r61668 = r61657 <= r61667;
        double r61669 = 1.0;
        double r61670 = 2.0;
        double r61671 = r61670 * r61663;
        double r61672 = -r61657;
        double r61673 = r61657 * r61657;
        double r61674 = 4.0;
        double r61675 = r61674 * r61663;
        double r61676 = r61675 * r61661;
        double r61677 = r61673 - r61676;
        double r61678 = sqrt(r61677);
        double r61679 = r61672 + r61678;
        double r61680 = r61671 / r61679;
        double r61681 = r61669 / r61680;
        double r61682 = 1.5059650565626715e+74;
        bool r61683 = r61657 <= r61682;
        double r61684 = 0.0;
        double r61685 = r61684 + r61676;
        double r61686 = r61672 - r61678;
        double r61687 = r61685 / r61686;
        double r61688 = r61669 / r61671;
        double r61689 = r61687 * r61688;
        double r61690 = -1.0;
        double r61691 = r61690 * r61662;
        double r61692 = r61683 ? r61689 : r61691;
        double r61693 = r61668 ? r61681 : r61692;
        double r61694 = r61659 ? r61666 : r61693;
        return r61694;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -1.361733299857302e+105

    1. Initial program 48.6

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.361733299857302e+105 < b < 6.89680782654711e-222

    1. Initial program 9.6

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

    3. Applied clear-num9.8

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

    if 6.89680782654711e-222 < b < 1.5059650565626715e+74

    1. Initial program 33.9

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

    3. Applied flip-+34.0

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

    4. Simplified16.5

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

    6. Applied div-inv16.5

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

    if 1.5059650565626715e+74 < b

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.361733299857302083043096878302889042354 \cdot 10^{105}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.896807826547109987464579044865965135876 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{elif}\;b \le 1.505965056562671542641677182626911662098 \cdot 10^{74}:\\ \;\;\;\;\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))