Average Error: 28.1 → 16.5
Time: 21.5s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 208.2161789220425021085247863084077835083:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 208.2161789220425021085247863084077835083:\\
\;\;\;\;\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)}}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1575930 = b;
        double r1575931 = -r1575930;
        double r1575932 = r1575930 * r1575930;
        double r1575933 = 4.0;
        double r1575934 = a;
        double r1575935 = r1575933 * r1575934;
        double r1575936 = c;
        double r1575937 = r1575935 * r1575936;
        double r1575938 = r1575932 - r1575937;
        double r1575939 = sqrt(r1575938);
        double r1575940 = r1575931 + r1575939;
        double r1575941 = 2.0;
        double r1575942 = r1575941 * r1575934;
        double r1575943 = r1575940 / r1575942;
        return r1575943;
}
↓
double f(double a, double b, double c) {
        double r1575944 = b;
        double r1575945 = 208.2161789220425;
        bool r1575946 = r1575944 <= r1575945;
        double r1575947 = r1575944 * r1575944;
        double r1575948 = 4.0;
        double r1575949 = c;
        double r1575950 = a;
        double r1575951 = r1575949 * r1575950;
        double r1575952 = r1575948 * r1575951;
        double r1575953 = r1575947 - r1575952;
        double r1575954 = r1575953 - r1575947;
        double r1575955 = sqrt(r1575953);
        double r1575956 = r1575944 + r1575955;
        double r1575957 = r1575954 / r1575956;
        double r1575958 = 2.0;
        double r1575959 = r1575957 / r1575958;
        double r1575960 = r1575959 / r1575950;
        double r1575961 = r1575949 / r1575944;
        double r1575962 = -1.0;
        double r1575963 = r1575961 * r1575962;
        double r1575964 = r1575946 ? r1575960 : r1575963;
        return r1575964;
}

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 < 208.2161789220425

    1. Initial program 15.4

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

    2. Simplified15.4

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

    4. Applied flip--15.4

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

    5. Simplified14.4

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

    if 208.2161789220425 < b

    1. Initial program 34.7

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

    2. Simplified34.7

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

    3. Taylor expanded around inf 17.6

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2}}{a}\]
    4. Using strategy rm

    5. Applied add-sqr-sqrt17.7

      \[\leadsto \frac{\frac{-2 \cdot \frac{a \cdot c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}}{2}}{a}\]

    6. Applied associate-/r*17.7

      \[\leadsto \frac{\frac{-2 \cdot \color{blue}{\frac{\frac{a \cdot c}{\sqrt{b}}}{\sqrt{b}}}}{2}}{a}\]

    7. Taylor expanded around 0 17.5

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

  4. Final simplification16.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 208.2161789220425021085247863084077835083:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right) - b \cdot b}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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) (* (* 4.0 a) c)))) (* 2.0 a)))