Average Error: 34.1 → 9.9
Time: 15.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r40944 = b;
        double r40945 = -r40944;
        double r40946 = r40944 * r40944;
        double r40947 = 4.0;
        double r40948 = a;
        double r40949 = r40947 * r40948;
        double r40950 = c;
        double r40951 = r40949 * r40950;
        double r40952 = r40946 - r40951;
        double r40953 = sqrt(r40952);
        double r40954 = r40945 + r40953;
        double r40955 = 2.0;
        double r40956 = r40955 * r40948;
        double r40957 = r40954 / r40956;
        return r40957;
}
↓
double f(double a, double b, double c) {
        double r40958 = b;
        double r40959 = -1.5501620157466267e+150;
        bool r40960 = r40958 <= r40959;
        double r40961 = 1.0;
        double r40962 = c;
        double r40963 = r40962 / r40958;
        double r40964 = a;
        double r40965 = r40958 / r40964;
        double r40966 = r40963 - r40965;
        double r40967 = r40961 * r40966;
        double r40968 = 1.611450844781215e-34;
        bool r40969 = r40958 <= r40968;
        double r40970 = 1.0;
        double r40971 = 2.0;
        double r40972 = r40971 * r40964;
        double r40973 = r40958 * r40958;
        double r40974 = 4.0;
        double r40975 = r40974 * r40964;
        double r40976 = r40975 * r40962;
        double r40977 = r40973 - r40976;
        double r40978 = sqrt(r40977);
        double r40979 = r40978 - r40958;
        double r40980 = r40972 / r40979;
        double r40981 = r40970 / r40980;
        double r40982 = -1.0;
        double r40983 = r40982 * r40963;
        double r40984 = r40969 ? r40981 : r40983;
        double r40985 = r40960 ? r40967 : r40984;
        return r40985;
}

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 < -1.5501620157466267e+150

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 1.7

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified1.7

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

    if -1.5501620157466267e+150 < b < 1.611450844781215e-34

    1. Initial program 13.6

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

    3. Applied clear-num13.7

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

    4. Simplified13.7

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

    if 1.611450844781215e-34 < b

    1. Initial program 55.0

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

    2. Taylor expanded around inf 7.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))