Average Error: 34.2 → 7.1
Time: 4.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -9.501658503658959535381027650257422342525 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.203463322495104127662247988126463798444 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -9.501658503658959535381027650257422342525 \cdot 10^{153}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.203463322495104127662247988126463798444 \cdot 10^{-228}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r67139 = b_2;
        double r67140 = -r67139;
        double r67141 = r67139 * r67139;
        double r67142 = a;
        double r67143 = c;
        double r67144 = r67142 * r67143;
        double r67145 = r67141 - r67144;
        double r67146 = sqrt(r67145);
        double r67147 = r67140 - r67146;
        double r67148 = r67147 / r67142;
        return r67148;
}
↓
double f(double a, double b_2, double c) {
        double r67149 = b_2;
        double r67150 = -9.50165850365896e+153;
        bool r67151 = r67149 <= r67150;
        double r67152 = -0.5;
        double r67153 = c;
        double r67154 = r67153 / r67149;
        double r67155 = r67152 * r67154;
        double r67156 = 3.203463322495104e-228;
        bool r67157 = r67149 <= r67156;
        double r67158 = 1.0;
        double r67159 = a;
        double r67160 = r67158 / r67159;
        double r67161 = 0.0;
        double r67162 = r67160 * r67161;
        double r67163 = r67158 * r67153;
        double r67164 = r67162 + r67163;
        double r67165 = -r67149;
        double r67166 = r67149 * r67149;
        double r67167 = r67159 * r67153;
        double r67168 = r67166 - r67167;
        double r67169 = sqrt(r67168);
        double r67170 = r67165 + r67169;
        double r67171 = r67164 / r67170;
        double r67172 = 6.568668442325333e+48;
        bool r67173 = r67149 <= r67172;
        double r67174 = r67165 - r67169;
        double r67175 = r67174 * r67160;
        double r67176 = 0.5;
        double r67177 = r67176 * r67154;
        double r67178 = 2.0;
        double r67179 = r67149 / r67159;
        double r67180 = r67178 * r67179;
        double r67181 = r67177 - r67180;
        double r67182 = r67173 ? r67175 : r67181;
        double r67183 = r67157 ? r67171 : r67182;
        double r67184 = r67151 ? r67155 : r67183;
        return r67184;
}

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 4 regimes

  2. if b_2 < -9.50165850365896e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 1.6

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

    if -9.50165850365896e+153 < b_2 < 3.203463322495104e-228

    1. Initial program 31.4

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

    3. Applied div-inv31.5

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

    5. Applied flip--31.5

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

    6. Applied associate-*l/31.6

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

    7. Simplified14.7

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

    9. Applied *-un-lft-identity14.7

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

    10. Applied *-un-lft-identity14.7

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

    11. Applied times-frac14.7

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

    12. Applied associate-*l*14.7

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

    13. Simplified9.2

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

    if 3.203463322495104e-228 < b_2 < 6.568668442325333e+48

    1. Initial program 9.1

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

    3. Applied div-inv9.2

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

    if 6.568668442325333e+48 < b_2

    1. Initial program 39.1

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

    2. Taylor expanded around inf 5.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -9.501658503658959535381027650257422342525 \cdot 10^{153}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.203463322495104127662247988126463798444 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot 0 + 1 \cdot c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))