Average Error: 34.2 → 8.6
Time: 7.7s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.14194017547317126 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.5120874391809866 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.0231735748307204843:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.6383712677255495 \cdot 10^{30}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.29266152734135 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.14194017547317126 \cdot 10^{130}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -1.5120874391809866 \cdot 10^{-204}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 0.0231735748307204843:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\

\mathbf{elif}\;b_2 \le 4.6383712677255495 \cdot 10^{30}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.29266152734135 \cdot 10^{122}:\\
\;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r28182 = b_2;
        double r28183 = -r28182;
        double r28184 = r28182 * r28182;
        double r28185 = a;
        double r28186 = c;
        double r28187 = r28185 * r28186;
        double r28188 = r28184 - r28187;
        double r28189 = sqrt(r28188);
        double r28190 = r28183 + r28189;
        double r28191 = r28190 / r28185;
        return r28191;
}
↓
double f(double a, double b_2, double c) {
        double r28192 = b_2;
        double r28193 = -2.1419401754731713e+130;
        bool r28194 = r28192 <= r28193;
        double r28195 = 0.5;
        double r28196 = c;
        double r28197 = r28196 / r28192;
        double r28198 = r28195 * r28197;
        double r28199 = 2.0;
        double r28200 = a;
        double r28201 = r28192 / r28200;
        double r28202 = r28199 * r28201;
        double r28203 = r28198 - r28202;
        double r28204 = -1.5120874391809866e-204;
        bool r28205 = r28192 <= r28204;
        double r28206 = -r28192;
        double r28207 = r28192 * r28192;
        double r28208 = r28200 * r28196;
        double r28209 = r28207 - r28208;
        double r28210 = sqrt(r28209);
        double r28211 = r28206 + r28210;
        double r28212 = 1.0;
        double r28213 = r28212 / r28200;
        double r28214 = r28211 * r28213;
        double r28215 = 0.023173574830720484;
        bool r28216 = r28192 <= r28215;
        double r28217 = r28206 - r28210;
        double r28218 = r28217 / r28200;
        double r28219 = r28218 / r28196;
        double r28220 = r28212 / r28219;
        double r28221 = r28220 / r28200;
        double r28222 = 4.6383712677255495e+30;
        bool r28223 = r28192 <= r28222;
        double r28224 = -0.5;
        double r28225 = r28224 * r28197;
        double r28226 = 2.292661527341346e+122;
        bool r28227 = r28192 <= r28226;
        double r28228 = 0.0;
        double r28229 = r28228 + r28208;
        double r28230 = r28229 / r28217;
        double r28231 = r28230 / r28200;
        double r28232 = r28227 ? r28231 : r28225;
        double r28233 = r28223 ? r28225 : r28232;
        double r28234 = r28216 ? r28221 : r28233;
        double r28235 = r28205 ? r28214 : r28234;
        double r28236 = r28194 ? r28203 : r28235;
        return r28236;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 5 regimes

  2. if b_2 < -2.1419401754731713e+130

    1. Initial program 57.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 3.2

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    if -2.1419401754731713e+130 < b_2 < -1.5120874391809866e-204

    1. Initial program 7.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv7.2

      \[\leadsto \color{blue}{\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if -1.5120874391809866e-204 < b_2 < 0.023173574830720484

    1. Initial program 22.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+22.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified17.0

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied clear-num17.0

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{0 + a \cdot c}}}}{a}\]

    7. Simplified14.6

      \[\leadsto \frac{\frac{1}{\color{blue}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}}{a}\]

    if 0.023173574830720484 < b_2 < 4.6383712677255495e+30 or 2.292661527341346e+122 < b_2

    1. Initial program 59.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.1

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if 4.6383712677255495e+30 < b_2 < 2.292661527341346e+122

    1. Initial program 47.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip-+47.1

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified13.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.14194017547317126 \cdot 10^{130}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -1.5120874391809866 \cdot 10^{-204}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 0.0231735748307204843:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}{c}}}{a}\\ \mathbf{elif}\;b_2 \le 4.6383712677255495 \cdot 10^{30}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.29266152734135 \cdot 10^{122}:\\ \;\;\;\;\frac{\frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))