Average Error: 34.2 → 10.0
Time: 4.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -5.2389466313579672 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r49193 = b;
        double r49194 = -r49193;
        double r49195 = r49193 * r49193;
        double r49196 = 4.0;
        double r49197 = a;
        double r49198 = r49196 * r49197;
        double r49199 = c;
        double r49200 = r49198 * r49199;
        double r49201 = r49195 - r49200;
        double r49202 = sqrt(r49201);
        double r49203 = r49194 + r49202;
        double r49204 = 2.0;
        double r49205 = r49204 * r49197;
        double r49206 = r49203 / r49205;
        return r49206;
}
↓
double f(double a, double b, double c) {
        double r49207 = b;
        double r49208 = -5.238946631357967e+127;
        bool r49209 = r49207 <= r49208;
        double r49210 = 1.0;
        double r49211 = c;
        double r49212 = r49211 / r49207;
        double r49213 = a;
        double r49214 = r49207 / r49213;
        double r49215 = r49212 - r49214;
        double r49216 = r49210 * r49215;
        double r49217 = 1.667046824505827e-85;
        bool r49218 = r49207 <= r49217;
        double r49219 = 1.0;
        double r49220 = 2.0;
        double r49221 = r49220 * r49213;
        double r49222 = -r49207;
        double r49223 = r49207 * r49207;
        double r49224 = 4.0;
        double r49225 = r49224 * r49213;
        double r49226 = r49225 * r49211;
        double r49227 = r49223 - r49226;
        double r49228 = sqrt(r49227);
        double r49229 = r49222 + r49228;
        double r49230 = r49221 / r49229;
        double r49231 = r49219 / r49230;
        double r49232 = -1.0;
        double r49233 = r49232 * r49212;
        double r49234 = r49218 ? r49231 : r49233;
        double r49235 = r49209 ? r49216 : r49234;
        return r49235;
}

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 < -5.238946631357967e+127

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -5.238946631357967e+127 < b < 1.667046824505827e-85

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    if 1.667046824505827e-85 < b

    1. Initial program 52.8

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

    2. Taylor expanded around inf 9.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.6670468245058271 \cdot 10^{-85}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))