Average Error: 34.2 → 8.6
Time: 5.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.01140706391993812 \cdot 10^{-189}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 4.94451211601336354 \cdot 10^{45}:\\ \;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le 3.01140706391993812 \cdot 10^{-189}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 4.94451211601336354 \cdot 10^{45}:\\
\;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r17224 = b_2;
        double r17225 = -r17224;
        double r17226 = r17224 * r17224;
        double r17227 = a;
        double r17228 = c;
        double r17229 = r17227 * r17228;
        double r17230 = r17226 - r17229;
        double r17231 = sqrt(r17230);
        double r17232 = r17225 + r17231;
        double r17233 = r17232 / r17227;
        return r17233;
}
↓
double f(double a, double b_2, double c) {
        double r17234 = b_2;
        double r17235 = -1.0524969308880696e+141;
        bool r17236 = r17234 <= r17235;
        double r17237 = 0.5;
        double r17238 = c;
        double r17239 = r17238 / r17234;
        double r17240 = r17237 * r17239;
        double r17241 = 2.0;
        double r17242 = a;
        double r17243 = r17234 / r17242;
        double r17244 = r17241 * r17243;
        double r17245 = r17240 - r17244;
        double r17246 = 3.011407063919938e-189;
        bool r17247 = r17234 <= r17246;
        double r17248 = -r17234;
        double r17249 = r17234 * r17234;
        double r17250 = r17242 * r17238;
        double r17251 = r17249 - r17250;
        double r17252 = sqrt(r17251);
        double r17253 = r17248 + r17252;
        double r17254 = 1.0;
        double r17255 = r17254 / r17242;
        double r17256 = r17253 * r17255;
        double r17257 = 4.9445121160133635e+45;
        bool r17258 = r17234 <= r17257;
        double r17259 = r17242 / r17250;
        double r17260 = r17248 - r17252;
        double r17261 = r17259 * r17260;
        double r17262 = r17254 / r17261;
        double r17263 = -0.5;
        double r17264 = r17263 * r17239;
        double r17265 = r17258 ? r17262 : r17264;
        double r17266 = r17247 ? r17256 : r17265;
        double r17267 = r17236 ? r17245 : r17266;
        return r17267;
}

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 < -1.0524969308880696e+141

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 2.4

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

    if -1.0524969308880696e+141 < b_2 < 3.011407063919938e-189

    1. Initial program 10.5

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

    3. Applied div-inv10.7

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

    if 3.011407063919938e-189 < b_2 < 4.9445121160133635e+45

    1. Initial program 32.3

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

    3. Applied flip-+32.4

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

    4. Simplified16.5

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

    6. Applied *-un-lft-identity16.5

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

    7. Applied *-un-lft-identity16.5

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

    8. Applied times-frac16.5

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

    9. Applied associate-/l*16.6

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

    10. Simplified16.3

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

    if 4.9445121160133635e+45 < b_2

    1. Initial program 56.8

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

    2. Taylor expanded around inf 3.7

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.05249693088806959 \cdot 10^{141}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 3.01140706391993812 \cdot 10^{-189}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 4.94451211601336354 \cdot 10^{45}:\\ \;\;\;\;\frac{1}{\frac{a}{a \cdot c} \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))