Average Error: 33.6 → 9.7
Time: 27.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -4.694684309811035 \cdot 10^{+121}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1504917 = b;
        double r1504918 = -r1504917;
        double r1504919 = r1504917 * r1504917;
        double r1504920 = 4.0;
        double r1504921 = a;
        double r1504922 = r1504920 * r1504921;
        double r1504923 = c;
        double r1504924 = r1504922 * r1504923;
        double r1504925 = r1504919 - r1504924;
        double r1504926 = sqrt(r1504925);
        double r1504927 = r1504918 + r1504926;
        double r1504928 = 2.0;
        double r1504929 = r1504928 * r1504921;
        double r1504930 = r1504927 / r1504929;
        return r1504930;
}
↓
double f(double a, double b, double c) {
        double r1504931 = b;
        double r1504932 = -4.694684309811035e+121;
        bool r1504933 = r1504931 <= r1504932;
        double r1504934 = c;
        double r1504935 = r1504934 / r1504931;
        double r1504936 = a;
        double r1504937 = r1504931 / r1504936;
        double r1504938 = r1504935 - r1504937;
        double r1504939 = 2.0;
        double r1504940 = r1504938 * r1504939;
        double r1504941 = r1504940 / r1504939;
        double r1504942 = 4.6659701943749105e-84;
        bool r1504943 = r1504931 <= r1504942;
        double r1504944 = 1.0;
        double r1504945 = r1504944 / r1504936;
        double r1504946 = r1504931 * r1504931;
        double r1504947 = 4.0;
        double r1504948 = r1504934 * r1504936;
        double r1504949 = r1504947 * r1504948;
        double r1504950 = r1504946 - r1504949;
        double r1504951 = sqrt(r1504950);
        double r1504952 = r1504951 - r1504931;
        double r1504953 = r1504945 * r1504952;
        double r1504954 = r1504953 / r1504939;
        double r1504955 = -2.0;
        double r1504956 = r1504935 * r1504955;
        double r1504957 = r1504956 / r1504939;
        double r1504958 = r1504943 ? r1504954 : r1504957;
        double r1504959 = r1504933 ? r1504941 : r1504958;
        return r1504959;
}

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 3 regimes

  2. if b < -4.694684309811035e+121

    1. Initial program 49.8

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

    2. Simplified49.8

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

    4. Applied div-inv49.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right) \cdot \frac{1}{a}}}{2}\]

    5. Taylor expanded around -inf 2.6

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

    6. Simplified2.6

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

    if -4.694684309811035e+121 < b < 4.6659701943749105e-84

    1. Initial program 12.2

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

    2. Simplified12.2

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

    4. Applied div-inv12.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b\right) \cdot \frac{1}{a}}}{2}\]

    if 4.6659701943749105e-84 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.694684309811035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.6659701943749105 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))