Average Error: 34.2 → 6.6
Time: 5.6s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5254699676315931 \cdot 10^{122}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.27850456875614525 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.23248892134249817 \cdot 10^{79}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.5254699676315931 \cdot 10^{122}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\mathbf{elif}\;b_2 \le -6.27850456875614525 \cdot 10^{-182}:\\
\;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

\mathbf{elif}\;b_2 \le 5.23248892134249817 \cdot 10^{79}:\\
\;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r19368 = b_2;
        double r19369 = -r19368;
        double r19370 = r19368 * r19368;
        double r19371 = a;
        double r19372 = c;
        double r19373 = r19371 * r19372;
        double r19374 = r19370 - r19373;
        double r19375 = sqrt(r19374);
        double r19376 = r19369 + r19375;
        double r19377 = r19376 / r19371;
        return r19377;
}
↓
double f(double a, double b_2, double c) {
        double r19378 = b_2;
        double r19379 = -1.5254699676315931e+122;
        bool r19380 = r19378 <= r19379;
        double r19381 = 0.5;
        double r19382 = c;
        double r19383 = r19382 / r19378;
        double r19384 = r19381 * r19383;
        double r19385 = 2.0;
        double r19386 = a;
        double r19387 = r19378 / r19386;
        double r19388 = r19385 * r19387;
        double r19389 = r19384 - r19388;
        double r19390 = -6.278504568756145e-182;
        bool r19391 = r19378 <= r19390;
        double r19392 = -r19378;
        double r19393 = r19378 * r19378;
        double r19394 = r19386 * r19382;
        double r19395 = r19393 - r19394;
        double r19396 = sqrt(r19395);
        double r19397 = r19392 + r19396;
        double r19398 = 1.0;
        double r19399 = r19398 / r19386;
        double r19400 = r19397 * r19399;
        double r19401 = 5.232488921342498e+79;
        bool r19402 = r19378 <= r19401;
        double r19403 = r19392 - r19396;
        double r19404 = r19398 * r19403;
        double r19405 = r19404 / r19382;
        double r19406 = r19398 / r19405;
        double r19407 = -0.5;
        double r19408 = r19407 * r19383;
        double r19409 = r19402 ? r19406 : r19408;
        double r19410 = r19391 ? r19400 : r19409;
        double r19411 = r19380 ? r19389 : r19410;
        return r19411;
}

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 < -1.5254699676315931e+122

    1. Initial program 52.5

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

    2. Taylor expanded around -inf 2.7

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

    if -1.5254699676315931e+122 < b_2 < -6.278504568756145e-182

    1. Initial program 7.0

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

    3. Applied div-inv7.2

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

    if -6.278504568756145e-182 < b_2 < 5.232488921342498e+79

    1. Initial program 27.8

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

    3. Applied flip-+27.9

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

    4. Simplified16.1

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

    6. Applied *-un-lft-identity16.1

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

    7. Applied *-un-lft-identity16.1

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

    8. Applied times-frac16.1

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

    9. Simplified16.1

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

    10. Simplified14.1

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

    12. Applied clear-num14.1

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

    13. Simplified10.4

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

    if 5.232488921342498e+79 < b_2

    1. Initial program 58.7

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5254699676315931 \cdot 10^{122}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.27850456875614525 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.23248892134249817 \cdot 10^{79}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))