Average Error: 28.8 → 16.1
Time: 15.5s
Precision: 64
\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 1083.7283358723973:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r662779 = b;
        double r662780 = -r662779;
        double r662781 = r662779 * r662779;
        double r662782 = 4.0;
        double r662783 = a;
        double r662784 = r662782 * r662783;
        double r662785 = c;
        double r662786 = r662784 * r662785;
        double r662787 = r662781 - r662786;
        double r662788 = sqrt(r662787);
        double r662789 = r662780 + r662788;
        double r662790 = 2.0;
        double r662791 = r662790 * r662783;
        double r662792 = r662789 / r662791;
        return r662792;
}
↓
double f(double a, double b, double c) {
        double r662793 = b;
        double r662794 = 1083.7283358723973;
        bool r662795 = r662793 <= r662794;
        double r662796 = r662793 * r662793;
        double r662797 = 4.0;
        double r662798 = c;
        double r662799 = a;
        double r662800 = r662798 * r662799;
        double r662801 = r662797 * r662800;
        double r662802 = r662796 - r662801;
        double r662803 = r662802 - r662796;
        double r662804 = sqrt(r662802);
        double r662805 = r662793 + r662804;
        double r662806 = r662803 / r662805;
        double r662807 = r662806 / r662799;
        double r662808 = 2.0;
        double r662809 = r662807 / r662808;
        double r662810 = r662798 / r662793;
        double r662811 = -2.0;
        double r662812 = r662810 * r662811;
        double r662813 = r662812 / r662808;
        double r662814 = r662795 ? r662809 : r662813;
        return r662814;
}

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 < 1083.7283358723973

    1. Initial program 17.4

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

    2. Simplified17.4

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

    4. Applied flip--17.4

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

    5. Simplified16.4

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

    if 1083.7283358723973 < b

    1. Initial program 36.7

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

    2. Simplified36.7

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

    3. Taylor expanded around inf 15.9

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1083.7283358723973:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :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)))