Average Error: 34.3 → 6.6
Time: 15.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.353411485092052314829631610929360473517 \cdot 10^{-283}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.608284405260305560018146283533738265178 \cdot 10^{78}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.353411485092052314829631610929360473517 \cdot 10^{-283}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.608284405260305560018146283533738265178 \cdot 10^{78}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r35565 = b;
        double r35566 = -r35565;
        double r35567 = r35565 * r35565;
        double r35568 = 4.0;
        double r35569 = a;
        double r35570 = r35568 * r35569;
        double r35571 = c;
        double r35572 = r35570 * r35571;
        double r35573 = r35567 - r35572;
        double r35574 = sqrt(r35573);
        double r35575 = r35566 + r35574;
        double r35576 = 2.0;
        double r35577 = r35576 * r35569;
        double r35578 = r35575 / r35577;
        return r35578;
}
↓
double f(double a, double b, double c) {
        double r35579 = b;
        double r35580 = -1.569310777886352e+111;
        bool r35581 = r35579 <= r35580;
        double r35582 = 1.0;
        double r35583 = c;
        double r35584 = r35583 / r35579;
        double r35585 = a;
        double r35586 = r35579 / r35585;
        double r35587 = r35584 - r35586;
        double r35588 = r35582 * r35587;
        double r35589 = -3.3534114850920523e-283;
        bool r35590 = r35579 <= r35589;
        double r35591 = -r35579;
        double r35592 = r35579 * r35579;
        double r35593 = 4.0;
        double r35594 = r35593 * r35585;
        double r35595 = r35594 * r35583;
        double r35596 = r35592 - r35595;
        double r35597 = sqrt(r35596);
        double r35598 = r35591 + r35597;
        double r35599 = 1.0;
        double r35600 = 2.0;
        double r35601 = r35600 * r35585;
        double r35602 = r35599 / r35601;
        double r35603 = r35598 * r35602;
        double r35604 = 1.6082844052603056e+78;
        bool r35605 = r35579 <= r35604;
        double r35606 = r35600 * r35583;
        double r35607 = r35591 - r35597;
        double r35608 = r35606 / r35607;
        double r35609 = -1.0;
        double r35610 = r35609 * r35584;
        double r35611 = r35605 ? r35608 : r35610;
        double r35612 = r35590 ? r35603 : r35611;
        double r35613 = r35581 ? r35588 : r35612;
        return r35613;
}

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 < -1.569310777886352e+111

    1. Initial program 50.4

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

    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 -1.569310777886352e+111 < b < -3.3534114850920523e-283

    1. Initial program 8.4

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

    3. Applied div-inv8.6

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

    if -3.3534114850920523e-283 < b < 1.6082844052603056e+78

    1. Initial program 30.3

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

    3. Applied div-inv30.3

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

    5. Applied flip-+30.4

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

    6. Applied associate-*l/30.5

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

    7. Simplified15.7

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

    8. Taylor expanded around 0 9.1

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

    if 1.6082844052603056e+78 < b

    1. Initial program 58.7

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

    2. Taylor expanded around inf 3.2

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.353411485092052314829631610929360473517 \cdot 10^{-283}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.608284405260305560018146283533738265178 \cdot 10^{78}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))