Average Error: 34.5 → 6.8
Time: 4.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.96852488717399238 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.96852488717399238 \cdot 10^{152}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\
\;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{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 r16227 = b_2;
        double r16228 = -r16227;
        double r16229 = r16227 * r16227;
        double r16230 = a;
        double r16231 = c;
        double r16232 = r16230 * r16231;
        double r16233 = r16229 - r16232;
        double r16234 = sqrt(r16233);
        double r16235 = r16228 - r16234;
        double r16236 = r16235 / r16230;
        return r16236;
}
↓
double f(double a, double b_2, double c) {
        double r16237 = b_2;
        double r16238 = -6.968524887173992e+152;
        bool r16239 = r16237 <= r16238;
        double r16240 = -0.5;
        double r16241 = c;
        double r16242 = r16241 / r16237;
        double r16243 = r16240 * r16242;
        double r16244 = -7.593271511213179e-252;
        bool r16245 = r16237 <= r16244;
        double r16246 = r16237 * r16237;
        double r16247 = a;
        double r16248 = r16247 * r16241;
        double r16249 = r16246 - r16248;
        double r16250 = sqrt(r16249);
        double r16251 = r16250 - r16237;
        double r16252 = r16241 / r16251;
        double r16253 = 9.965676396086742e+45;
        bool r16254 = r16237 <= r16253;
        double r16255 = -r16237;
        double r16256 = r16255 / r16247;
        double r16257 = r16250 / r16247;
        double r16258 = r16256 - r16257;
        double r16259 = 0.5;
        double r16260 = r16259 * r16242;
        double r16261 = 2.0;
        double r16262 = r16237 / r16247;
        double r16263 = r16261 * r16262;
        double r16264 = r16260 - r16263;
        double r16265 = r16254 ? r16258 : r16264;
        double r16266 = r16245 ? r16252 : r16265;
        double r16267 = r16239 ? r16243 : r16266;
        return r16267;
}

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 < -6.968524887173992e+152

    1. Initial program 63.9

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

    2. Taylor expanded around -inf 1.5

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

    if -6.968524887173992e+152 < b_2 < -7.593271511213179e-252

    1. Initial program 36.7

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

    3. Applied flip--36.7

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

    4. Simplified16.0

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

    5. Simplified16.0

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

    7. Applied clear-num16.2

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

    8. Simplified14.5

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

    10. Applied associate-/r*14.2

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

    11. Simplified7.6

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

    if -7.593271511213179e-252 < b_2 < 9.965676396086742e+45

    1. Initial program 10.6

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

    3. Applied div-sub10.6

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

    if 9.965676396086742e+45 < b_2

    1. Initial program 36.8

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

    2. Taylor expanded around inf 5.2

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.96852488717399238 \cdot 10^{152}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -7.5932715112131794 \cdot 10^{-252}:\\ \;\;\;\;\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 9.9656763960867421 \cdot 10^{45}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))