Average Error: 34.7 → 6.9
Time: 7.1s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.90883508250240445 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 3.67086091268017442 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.3202538172935113 \cdot 10^{68}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -8.90883508250240445 \cdot 10^{-161}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 3.67086091268017442 \cdot 10^{125}:\\
\;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r24332 = b_2;
        double r24333 = -r24332;
        double r24334 = r24332 * r24332;
        double r24335 = a;
        double r24336 = c;
        double r24337 = r24335 * r24336;
        double r24338 = r24334 - r24337;
        double r24339 = sqrt(r24338);
        double r24340 = r24333 + r24339;
        double r24341 = r24340 / r24335;
        return r24341;
}
↓
double f(double a, double b_2, double c) {
        double r24342 = b_2;
        double r24343 = -2.3202538172935113e+68;
        bool r24344 = r24342 <= r24343;
        double r24345 = 0.5;
        double r24346 = c;
        double r24347 = r24346 / r24342;
        double r24348 = r24345 * r24347;
        double r24349 = 2.0;
        double r24350 = a;
        double r24351 = r24342 / r24350;
        double r24352 = r24349 * r24351;
        double r24353 = r24348 - r24352;
        double r24354 = -8.908835082502404e-161;
        bool r24355 = r24342 <= r24354;
        double r24356 = -r24342;
        double r24357 = r24342 * r24342;
        double r24358 = r24350 * r24346;
        double r24359 = r24357 - r24358;
        double r24360 = sqrt(r24359);
        double r24361 = r24356 + r24360;
        double r24362 = 1.0;
        double r24363 = r24362 / r24350;
        double r24364 = r24361 * r24363;
        double r24365 = 3.6708609126801744e+125;
        bool r24366 = r24342 <= r24365;
        double r24367 = r24356 - r24360;
        double r24368 = r24362 / r24367;
        double r24369 = r24362 / r24346;
        double r24370 = r24368 / r24369;
        double r24371 = -0.5;
        double r24372 = r24371 * r24347;
        double r24373 = r24366 ? r24370 : r24372;
        double r24374 = r24355 ? r24364 : r24373;
        double r24375 = r24344 ? r24353 : r24374;
        return r24375;
}

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 < -2.3202538172935113e+68

    1. Initial program 40.7

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

    2. Taylor expanded around -inf 5.2

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

    if -2.3202538172935113e+68 < b_2 < -8.908835082502404e-161

    1. Initial program 6.3

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

    3. Applied div-inv6.5

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

    if -8.908835082502404e-161 < b_2 < 3.6708609126801744e+125

    1. Initial program 29.8

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

    3. Applied flip-+30.1

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

    4. Simplified16.7

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

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

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

    7. Applied associate-/r*16.7

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

    8. Simplified14.9

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

    10. Applied div-inv15.0

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

    11. Applied *-un-lft-identity15.0

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

    12. Applied times-frac16.8

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

    13. Applied associate-/l*15.8

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

    14. Simplified10.6

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

    if 3.6708609126801744e+125 < b_2

    1. Initial program 61.6

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

    2. Taylor expanded around inf 1.7

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -8.90883508250240445 \cdot 10^{-161}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 3.67086091268017442 \cdot 10^{125}:\\ \;\;\;\;\frac{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\frac{1}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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))