Average Error: 34.5 → 11.0
Time: 7.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r35137 = b;
        double r35138 = -r35137;
        double r35139 = r35137 * r35137;
        double r35140 = 4.0;
        double r35141 = a;
        double r35142 = r35140 * r35141;
        double r35143 = c;
        double r35144 = r35142 * r35143;
        double r35145 = r35139 - r35144;
        double r35146 = sqrt(r35145);
        double r35147 = r35138 + r35146;
        double r35148 = 2.0;
        double r35149 = r35148 * r35141;
        double r35150 = r35147 / r35149;
        return r35150;
}
↓
double f(double a, double b, double c) {
        double r35151 = b;
        double r35152 = -2.6152573735422387e+153;
        bool r35153 = r35151 <= r35152;
        double r35154 = 1.0;
        double r35155 = c;
        double r35156 = r35155 / r35151;
        double r35157 = a;
        double r35158 = r35151 / r35157;
        double r35159 = r35156 - r35158;
        double r35160 = r35154 * r35159;
        double r35161 = 1.3880700472259379e-143;
        bool r35162 = r35151 <= r35161;
        double r35163 = -r35151;
        double r35164 = r35151 * r35151;
        double r35165 = 4.0;
        double r35166 = r35165 * r35157;
        double r35167 = r35166 * r35155;
        double r35168 = r35164 - r35167;
        double r35169 = sqrt(r35168);
        double r35170 = r35163 + r35169;
        double r35171 = 2.0;
        double r35172 = r35171 * r35157;
        double r35173 = r35170 / r35172;
        double r35174 = -1.0;
        double r35175 = r35174 * r35156;
        double r35176 = r35162 ? r35173 : r35175;
        double r35177 = r35153 ? r35160 : r35176;
        return r35177;
}

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 < -2.6152573735422387e+153

    1. Initial program 63.8

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

    2. Taylor expanded around -inf 2.1

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

    3. Simplified2.1

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

    if -2.6152573735422387e+153 < b < 1.3880700472259379e-143

    1. Initial program 11.5

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

    if 1.3880700472259379e-143 < b

    1. Initial program 50.3

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

    2. Taylor expanded around inf 12.6

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.615257373542238721197930661559276546696 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.388070047225937856958905133202240499626 \cdot 10^{-143}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))