Average Error: 34.1 → 9.9
Time: 17.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r46572 = b;
        double r46573 = -r46572;
        double r46574 = r46572 * r46572;
        double r46575 = 4.0;
        double r46576 = a;
        double r46577 = r46575 * r46576;
        double r46578 = c;
        double r46579 = r46577 * r46578;
        double r46580 = r46574 - r46579;
        double r46581 = sqrt(r46580);
        double r46582 = r46573 + r46581;
        double r46583 = 2.0;
        double r46584 = r46583 * r46576;
        double r46585 = r46582 / r46584;
        return r46585;
}
↓
double f(double a, double b, double c) {
        double r46586 = b;
        double r46587 = -1.5501620157466267e+150;
        bool r46588 = r46586 <= r46587;
        double r46589 = 1.0;
        double r46590 = c;
        double r46591 = r46590 / r46586;
        double r46592 = a;
        double r46593 = r46586 / r46592;
        double r46594 = r46591 - r46593;
        double r46595 = r46589 * r46594;
        double r46596 = 1.611450844781215e-34;
        bool r46597 = r46586 <= r46596;
        double r46598 = 1.0;
        double r46599 = 2.0;
        double r46600 = r46599 * r46592;
        double r46601 = r46586 * r46586;
        double r46602 = 4.0;
        double r46603 = r46602 * r46592;
        double r46604 = r46603 * r46590;
        double r46605 = r46601 - r46604;
        double r46606 = sqrt(r46605);
        double r46607 = r46606 - r46586;
        double r46608 = r46600 / r46607;
        double r46609 = r46598 / r46608;
        double r46610 = -1.0;
        double r46611 = r46610 * r46591;
        double r46612 = r46597 ? r46609 : r46611;
        double r46613 = r46588 ? r46595 : r46612;
        return r46613;
}

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 3 regimes

  2. if b < -1.5501620157466267e+150

    1. Initial program 62.9

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

    2. Simplified62.9

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

    3. Taylor expanded around -inf 1.7

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

    4. Simplified1.7

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

    if -1.5501620157466267e+150 < b < 1.611450844781215e-34

    1. Initial program 13.6

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

    2. Simplified13.6

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

    4. Applied clear-num13.7

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

    if 1.611450844781215e-34 < b

    1. Initial program 55.0

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

    2. Simplified55.0

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

    3. Taylor expanded around inf 7.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))