Average Error: 33.9 → 7.2
Time: 6.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.2375225949334019 \cdot 10^{57}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.9220534958503673 \cdot 10^{-246}:\\ \;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.77017414835012383 \cdot 10^{70}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.2375225949334019 \cdot 10^{57}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.9220534958503673 \cdot 10^{-246}:\\
\;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\

\mathbf{elif}\;b_2 \le 1.77017414835012383 \cdot 10^{70}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r21372 = b_2;
        double r21373 = -r21372;
        double r21374 = r21372 * r21372;
        double r21375 = a;
        double r21376 = c;
        double r21377 = r21375 * r21376;
        double r21378 = r21374 - r21377;
        double r21379 = sqrt(r21378);
        double r21380 = r21373 - r21379;
        double r21381 = r21380 / r21375;
        return r21381;
}
↓
double f(double a, double b_2, double c) {
        double r21382 = b_2;
        double r21383 = -2.237522594933402e+57;
        bool r21384 = r21382 <= r21383;
        double r21385 = -0.5;
        double r21386 = c;
        double r21387 = r21386 / r21382;
        double r21388 = r21385 * r21387;
        double r21389 = 1.9220534958503673e-246;
        bool r21390 = r21382 <= r21389;
        double r21391 = 1.0;
        double r21392 = r21382 * r21382;
        double r21393 = a;
        double r21394 = r21393 * r21386;
        double r21395 = r21392 - r21394;
        double r21396 = sqrt(r21395);
        double r21397 = r21396 - r21382;
        double r21398 = r21386 / r21397;
        double r21399 = r21391 * r21398;
        double r21400 = 1.7701741483501238e+70;
        bool r21401 = r21382 <= r21400;
        double r21402 = -r21382;
        double r21403 = r21402 - r21396;
        double r21404 = r21391 / r21393;
        double r21405 = r21403 * r21404;
        double r21406 = -2.0;
        double r21407 = r21382 / r21393;
        double r21408 = r21406 * r21407;
        double r21409 = r21401 ? r21405 : r21408;
        double r21410 = r21390 ? r21399 : r21409;
        double r21411 = r21384 ? r21388 : r21410;
        return r21411;
}

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 < -2.237522594933402e+57

    1. Initial program 57.0

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

    2. Taylor expanded around -inf 3.8

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

    if -2.237522594933402e+57 < b_2 < 1.9220534958503673e-246

    1. Initial program 28.3

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

    3. Applied flip--28.3

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

    4. Simplified17.0

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

    5. Simplified17.0

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

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

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

    8. Applied associate-/r*17.0

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

    9. Simplified14.6

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

    11. Applied *-un-lft-identity14.6

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

    12. Applied *-un-lft-identity14.6

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

    13. Applied times-frac14.6

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

    14. Applied associate-/l*14.6

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

    15. Simplified10.6

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

    17. Applied *-un-lft-identity10.6

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

    18. Applied *-un-lft-identity10.6

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

    19. Applied times-frac10.6

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

    20. Applied add-sqr-sqrt10.6

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

    21. Applied times-frac10.6

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

    22. Simplified10.6

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

    23. Simplified10.4

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

    if 1.9220534958503673e-246 < b_2 < 1.7701741483501238e+70

    1. Initial program 8.4

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

    3. Applied div-inv8.6

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

    if 1.7701741483501238e+70 < b_2

    1. Initial program 41.6

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

    3. Applied flip--62.0

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

    4. Simplified61.2

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

    5. Simplified61.2

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

    7. Applied *-un-lft-identity61.2

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

    8. Applied associate-/r*61.2

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

    9. Simplified61.0

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

    10. Taylor expanded around 0 5.6

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

  4. Final simplification7.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.2375225949334019 \cdot 10^{57}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.9220534958503673 \cdot 10^{-246}:\\ \;\;\;\;1 \cdot \frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}\\ \mathbf{elif}\;b_2 \le 1.77017414835012383 \cdot 10^{70}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))