Average Error: 34.9 → 6.6
Time: 18.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.842745537862711243019551203235877534619 \cdot 10^{150}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.515787668275236497163507166686881482854 \cdot 10^{-293}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.340805540534955068075384182716868662315 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.842745537862711243019551203235877534619 \cdot 10^{150}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.515787668275236497163507166686881482854 \cdot 10^{-293}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 4.340805540534955068075384182716868662315 \cdot 10^{107}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

\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 r78210 = b_2;
        double r78211 = -r78210;
        double r78212 = r78210 * r78210;
        double r78213 = a;
        double r78214 = c;
        double r78215 = r78213 * r78214;
        double r78216 = r78212 - r78215;
        double r78217 = sqrt(r78216);
        double r78218 = r78211 - r78217;
        double r78219 = r78218 / r78213;
        return r78219;
}
↓
double f(double a, double b_2, double c) {
        double r78220 = b_2;
        double r78221 = -1.8427455378627112e+150;
        bool r78222 = r78220 <= r78221;
        double r78223 = -0.5;
        double r78224 = c;
        double r78225 = r78224 / r78220;
        double r78226 = r78223 * r78225;
        double r78227 = -2.5157876682752365e-293;
        bool r78228 = r78220 <= r78227;
        double r78229 = r78220 * r78220;
        double r78230 = a;
        double r78231 = r78230 * r78224;
        double r78232 = r78229 - r78231;
        double r78233 = sqrt(r78232);
        double r78234 = r78233 - r78220;
        double r78235 = r78224 / r78234;
        double r78236 = 4.340805540534955e+107;
        bool r78237 = r78220 <= r78236;
        double r78238 = 1.0;
        double r78239 = -r78220;
        double r78240 = r78239 - r78233;
        double r78241 = r78230 / r78240;
        double r78242 = r78238 / r78241;
        double r78243 = 0.5;
        double r78244 = r78243 * r78225;
        double r78245 = 2.0;
        double r78246 = r78220 / r78230;
        double r78247 = r78245 * r78246;
        double r78248 = r78244 - r78247;
        double r78249 = r78237 ? r78242 : r78248;
        double r78250 = r78228 ? r78235 : r78249;
        double r78251 = r78222 ? r78226 : r78250;
        return r78251;
}

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 < -1.8427455378627112e+150

    1. Initial program 63.7

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

    2. Taylor expanded around -inf 1.7

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

    if -1.8427455378627112e+150 < b_2 < -2.5157876682752365e-293

    1. Initial program 35.8

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

    3. Applied flip--35.8

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

    4. Simplified16.9

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

    5. Simplified16.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied div-inv17.0

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

    9. Applied *-un-lft-identity17.0

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

    10. Applied *-un-lft-identity17.0

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

    11. Applied times-frac17.0

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

    12. Applied associate-*l*17.0

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

    13. Simplified15.2

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

    14. Taylor expanded around 0 8.2

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

    if -2.5157876682752365e-293 < b_2 < 4.340805540534955e+107

    1. Initial program 9.4

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

    3. Applied clear-num9.5

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

    if 4.340805540534955e+107 < b_2

    1. Initial program 49.7

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.842745537862711243019551203235877534619 \cdot 10^{150}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.515787668275236497163507166686881482854 \cdot 10^{-293}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 4.340805540534955068075384182716868662315 \cdot 10^{107}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))