Average Error: 33.8 → 9.5
Time: 18.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.517926393801403 \cdot 10^{+142}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1920157 = b;
        double r1920158 = -r1920157;
        double r1920159 = r1920157 * r1920157;
        double r1920160 = 4.0;
        double r1920161 = a;
        double r1920162 = r1920160 * r1920161;
        double r1920163 = c;
        double r1920164 = r1920162 * r1920163;
        double r1920165 = r1920159 - r1920164;
        double r1920166 = sqrt(r1920165);
        double r1920167 = r1920158 + r1920166;
        double r1920168 = 2.0;
        double r1920169 = r1920168 * r1920161;
        double r1920170 = r1920167 / r1920169;
        return r1920170;
}
↓
double f(double a, double b, double c) {
        double r1920171 = b;
        double r1920172 = -5.517926393801403e+142;
        bool r1920173 = r1920171 <= r1920172;
        double r1920174 = c;
        double r1920175 = r1920174 / r1920171;
        double r1920176 = a;
        double r1920177 = r1920171 / r1920176;
        double r1920178 = r1920175 - r1920177;
        double r1920179 = 1.3635892865650846e-93;
        bool r1920180 = r1920171 <= r1920179;
        double r1920181 = -r1920171;
        double r1920182 = r1920171 * r1920171;
        double r1920183 = 4.0;
        double r1920184 = r1920183 * r1920176;
        double r1920185 = r1920174 * r1920184;
        double r1920186 = r1920182 - r1920185;
        double r1920187 = sqrt(r1920186);
        double r1920188 = r1920181 + r1920187;
        double r1920189 = 0.5;
        double r1920190 = r1920189 / r1920176;
        double r1920191 = r1920188 * r1920190;
        double r1920192 = -r1920175;
        double r1920193 = r1920180 ? r1920191 : r1920192;
        double r1920194 = r1920173 ? r1920178 : r1920193;
        return r1920194;
}

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 < -5.517926393801403e+142

    1. Initial program 56.5

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

    2. Taylor expanded around -inf 2.7

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

    if -5.517926393801403e+142 < b < 1.3635892865650846e-93

    1. Initial program 11.6

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

    3. Applied div-inv11.7

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

    4. Simplified11.7

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

    if 1.3635892865650846e-93 < b

    1. Initial program 52.4

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

    2. Taylor expanded around inf 9.1

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

    3. Simplified9.1

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))