Average Error: 34.1 → 6.7
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.223763057046510327568967152287533282505 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.319380566438366601816459280349243307141 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\ \mathbf{elif}\;b \le 1.458057835821772074616178333218437979276 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.223763057046510327568967152287533282505 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.319380566438366601816459280349243307141 \cdot 10^{-186}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\

\mathbf{elif}\;b \le 1.458057835821772074616178333218437979276 \cdot 10^{144}:\\
\;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\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 r61160 = b;
        double r61161 = -r61160;
        double r61162 = r61160 * r61160;
        double r61163 = 4.0;
        double r61164 = a;
        double r61165 = r61163 * r61164;
        double r61166 = c;
        double r61167 = r61165 * r61166;
        double r61168 = r61162 - r61167;
        double r61169 = sqrt(r61168);
        double r61170 = r61161 + r61169;
        double r61171 = 2.0;
        double r61172 = r61171 * r61164;
        double r61173 = r61170 / r61172;
        return r61173;
}
↓
double f(double a, double b, double c) {
        double r61174 = b;
        double r61175 = -2.2237630570465103e+109;
        bool r61176 = r61174 <= r61175;
        double r61177 = 1.0;
        double r61178 = c;
        double r61179 = r61178 / r61174;
        double r61180 = a;
        double r61181 = r61174 / r61180;
        double r61182 = r61179 - r61181;
        double r61183 = r61177 * r61182;
        double r61184 = -3.3193805664383666e-186;
        bool r61185 = r61174 <= r61184;
        double r61186 = -r61174;
        double r61187 = r61174 * r61174;
        double r61188 = 4.0;
        double r61189 = r61188 * r61180;
        double r61190 = r61189 * r61178;
        double r61191 = r61187 - r61190;
        double r61192 = sqrt(r61191);
        double r61193 = r61186 + r61192;
        double r61194 = sqrt(r61193);
        double r61195 = 2.0;
        double r61196 = r61194 / r61195;
        double r61197 = r61194 / r61180;
        double r61198 = r61196 * r61197;
        double r61199 = 1.458057835821772e+144;
        bool r61200 = r61174 <= r61199;
        double r61201 = 1.0;
        double r61202 = r61195 / r61188;
        double r61203 = r61201 / r61202;
        double r61204 = r61203 * r61178;
        double r61205 = r61186 - r61192;
        double r61206 = r61204 / r61205;
        double r61207 = -1.0;
        double r61208 = r61207 * r61179;
        double r61209 = r61200 ? r61206 : r61208;
        double r61210 = r61185 ? r61198 : r61209;
        double r61211 = r61176 ? r61183 : r61210;
        return r61211;
}

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 4 regimes

  2. if b < -2.2237630570465103e+109

    1. Initial program 48.6

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

    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 -2.2237630570465103e+109 < b < -3.3193805664383666e-186

    1. Initial program 6.9

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

    3. Applied add-sqr-sqrt7.3

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

    4. Applied times-frac7.3

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

    if -3.3193805664383666e-186 < b < 1.458057835821772e+144

    1. Initial program 31.3

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

    3. Applied flip-+31.5

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

    4. Simplified16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num16.3

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified15.3

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

    9. Applied times-frac15.3

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

    10. Simplified10.2

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

    12. Applied associate-/r*9.9

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

    13. Simplified9.8

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

    if 1.458057835821772e+144 < b

    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 1.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.223763057046510327568967152287533282505 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.319380566438366601816459280349243307141 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2} \cdot \frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a}\\ \mathbf{elif}\;b \le 1.458057835821772074616178333218437979276 \cdot 10^{144}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))