Average Error: 28.3 → 16.3
Time: 7.0s
Precision: 64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 904.955392750566375:\\
\;\;\;\;\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 r22631 = b;
        double r22632 = -r22631;
        double r22633 = r22631 * r22631;
        double r22634 = 4.0;
        double r22635 = a;
        double r22636 = r22634 * r22635;
        double r22637 = c;
        double r22638 = r22636 * r22637;
        double r22639 = r22633 - r22638;
        double r22640 = sqrt(r22639);
        double r22641 = r22632 + r22640;
        double r22642 = 2.0;
        double r22643 = r22642 * r22635;
        double r22644 = r22641 / r22643;
        return r22644;
}
↓
double f(double a, double b, double c) {
        double r22645 = b;
        double r22646 = 904.9553927505664;
        bool r22647 = r22645 <= r22646;
        double r22648 = 2.0;
        double r22649 = pow(r22645, r22648);
        double r22650 = 4.0;
        double r22651 = a;
        double r22652 = c;
        double r22653 = r22651 * r22652;
        double r22654 = r22650 * r22653;
        double r22655 = r22649 - r22654;
        double r22656 = r22655 - r22649;
        double r22657 = r22645 * r22645;
        double r22658 = r22650 * r22651;
        double r22659 = r22658 * r22652;
        double r22660 = r22657 - r22659;
        double r22661 = sqrt(r22660);
        double r22662 = r22661 + r22645;
        double r22663 = r22656 / r22662;
        double r22664 = 2.0;
        double r22665 = r22663 / r22664;
        double r22666 = r22665 / r22651;
        double r22667 = -1.0;
        double r22668 = r22652 / r22645;
        double r22669 = r22667 * r22668;
        double r22670 = r22647 ? r22666 : r22669;
        return r22670;
}

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 < 904.9553927505664

    1. Initial program 17.1

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

    2. Simplified17.1

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

    4. Applied flip--17.1

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

    5. Simplified16.0

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

    if 904.9553927505664 < b

    1. Initial program 36.0

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

    2. Simplified36.0

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

    3. Taylor expanded around inf 16.5

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

  4. Final simplification16.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 904.955392750566375:\\ \;\;\;\;\frac{\frac{\frac{\left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right) - {b}^{2}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))