Average Error: 28.7 → 17.5
Time: 23.6s
Precision: 64
\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 14.42119907524596733594535180600360035896:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right) + b \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot c}{b} \cdot -2\right) \cdot \frac{1}{2 \cdot a}\\ \end{array}\]
\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 14.42119907524596733594535180600360035896:\\
\;\;\;\;\frac{\frac{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right) + b \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{a \cdot c}{b} \cdot -2\right) \cdot \frac{1}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r1696150 = b;
        double r1696151 = -r1696150;
        double r1696152 = r1696150 * r1696150;
        double r1696153 = 4.0;
        double r1696154 = a;
        double r1696155 = r1696153 * r1696154;
        double r1696156 = c;
        double r1696157 = r1696155 * r1696156;
        double r1696158 = r1696152 - r1696157;
        double r1696159 = sqrt(r1696158);
        double r1696160 = r1696151 + r1696159;
        double r1696161 = 2.0;
        double r1696162 = r1696161 * r1696154;
        double r1696163 = r1696160 / r1696162;
        return r1696163;
}
↓
double f(double a, double b, double c) {
        double r1696164 = b;
        double r1696165 = 14.421199075245967;
        bool r1696166 = r1696164 <= r1696165;
        double r1696167 = r1696164 * r1696164;
        double r1696168 = 4.0;
        double r1696169 = a;
        double r1696170 = c;
        double r1696171 = r1696169 * r1696170;
        double r1696172 = r1696168 * r1696171;
        double r1696173 = r1696167 - r1696172;
        double r1696174 = sqrt(r1696173);
        double r1696175 = r1696173 * r1696174;
        double r1696176 = r1696167 * r1696164;
        double r1696177 = r1696175 - r1696176;
        double r1696178 = r1696167 + r1696173;
        double r1696179 = r1696164 * r1696174;
        double r1696180 = r1696178 + r1696179;
        double r1696181 = r1696177 / r1696180;
        double r1696182 = 2.0;
        double r1696183 = r1696182 * r1696169;
        double r1696184 = r1696181 / r1696183;
        double r1696185 = r1696171 / r1696164;
        double r1696186 = -2.0;
        double r1696187 = r1696185 * r1696186;
        double r1696188 = 1.0;
        double r1696189 = r1696188 / r1696183;
        double r1696190 = r1696187 * r1696189;
        double r1696191 = r1696166 ? r1696184 : r1696190;
        return r1696191;
}

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 < 14.421199075245967

    1. Initial program 13.9

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

    2. Simplified13.9

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

    4. Applied flip3--14.0

      \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{2 \cdot a}\]

    5. Simplified13.3

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

    6. Simplified13.3

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

    if 14.421199075245967 < b

    1. Initial program 33.2

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

    2. Simplified33.2

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

    3. Taylor expanded around inf 18.8

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

    5. Applied div-inv18.8

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

  4. Final simplification17.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 14.42119907524596733594535180600360035896:\\ \;\;\;\;\frac{\frac{\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - \left(b \cdot b\right) \cdot b}{\left(b \cdot b + \left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right) + b \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{a \cdot c}{b} \cdot -2\right) \cdot \frac{1}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(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)))