Average Error: 28.5 → 16.2
Time: 8.6s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3187.18097597923543:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 3187.18097597923543:\\
\;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r27579 = b;
        double r27580 = -r27579;
        double r27581 = r27579 * r27579;
        double r27582 = 4.0;
        double r27583 = a;
        double r27584 = r27582 * r27583;
        double r27585 = c;
        double r27586 = r27584 * r27585;
        double r27587 = r27581 - r27586;
        double r27588 = sqrt(r27587);
        double r27589 = r27580 + r27588;
        double r27590 = 2.0;
        double r27591 = r27590 * r27583;
        double r27592 = r27589 / r27591;
        return r27592;
}
↓
double f(double a, double b, double c) {
        double r27593 = b;
        double r27594 = 3187.1809759792354;
        bool r27595 = r27593 <= r27594;
        double r27596 = 2.0;
        double r27597 = pow(r27593, r27596);
        double r27598 = 4.0;
        double r27599 = a;
        double r27600 = c;
        double r27601 = r27599 * r27600;
        double r27602 = r27598 * r27601;
        double r27603 = r27597 - r27602;
        double r27604 = r27603 - r27597;
        double r27605 = r27593 * r27593;
        double r27606 = r27598 * r27599;
        double r27607 = r27606 * r27600;
        double r27608 = r27605 - r27607;
        double r27609 = sqrt(r27608);
        double r27610 = r27609 + r27593;
        double r27611 = r27604 / r27610;
        double r27612 = 2.0;
        double r27613 = r27611 / r27612;
        double r27614 = r27613 / r27599;
        double r27615 = -1.0;
        double r27616 = r27600 / r27593;
        double r27617 = r27615 * r27616;
        double r27618 = r27595 ? r27614 : r27617;
        return r27618;
}

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 2 regimes

  2. if b < 3187.1809759792354

    1. Initial program 18.2

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

    2. Simplified18.2

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

    4. Applied flip--18.2

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

    5. Simplified17.1

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

    if 3187.1809759792354 < b

    1. Initial program 37.2

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

    2. Simplified37.2

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

    3. Taylor expanded around inf 15.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3187.18097597923543:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))