Average Error: 34.2 → 6.8
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.17047858644702483 \cdot 10^{-264}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.7711811459025421 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{\frac{2}{4}}}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 7.17047858644702483 \cdot 10^{-264}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 3.7711811459025421 \cdot 10^{84}:\\
\;\;\;\;1 \cdot \frac{\frac{1}{\frac{2}{4}}}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r75369 = b;
        double r75370 = -r75369;
        double r75371 = r75369 * r75369;
        double r75372 = 4.0;
        double r75373 = a;
        double r75374 = c;
        double r75375 = r75373 * r75374;
        double r75376 = r75372 * r75375;
        double r75377 = r75371 - r75376;
        double r75378 = sqrt(r75377);
        double r75379 = r75370 + r75378;
        double r75380 = 2.0;
        double r75381 = r75380 * r75373;
        double r75382 = r75379 / r75381;
        return r75382;
}
↓
double f(double a, double b, double c) {
        double r75383 = b;
        double r75384 = -5.238946631357967e+127;
        bool r75385 = r75383 <= r75384;
        double r75386 = 1.0;
        double r75387 = c;
        double r75388 = r75387 / r75383;
        double r75389 = a;
        double r75390 = r75383 / r75389;
        double r75391 = r75388 - r75390;
        double r75392 = r75386 * r75391;
        double r75393 = 7.170478586447025e-264;
        bool r75394 = r75383 <= r75393;
        double r75395 = -r75383;
        double r75396 = r75383 * r75383;
        double r75397 = 4.0;
        double r75398 = r75389 * r75387;
        double r75399 = r75397 * r75398;
        double r75400 = r75396 - r75399;
        double r75401 = sqrt(r75400);
        double r75402 = r75395 + r75401;
        double r75403 = 2.0;
        double r75404 = r75403 * r75389;
        double r75405 = r75402 / r75404;
        double r75406 = 3.771181145902542e+84;
        bool r75407 = r75383 <= r75406;
        double r75408 = 1.0;
        double r75409 = r75403 / r75397;
        double r75410 = r75408 / r75409;
        double r75411 = r75395 - r75401;
        double r75412 = r75411 / r75387;
        double r75413 = r75410 / r75412;
        double r75414 = r75408 * r75413;
        double r75415 = -1.0;
        double r75416 = r75415 * r75388;
        double r75417 = r75407 ? r75414 : r75416;
        double r75418 = r75394 ? r75405 : r75417;
        double r75419 = r75385 ? r75392 : r75418;
        return r75419;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.2
Target21.6
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.238946631357967e+127

    1. Initial program 54.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 3.3

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

    3. Simplified3.3

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

    if -5.238946631357967e+127 < b < 7.170478586447025e-264

    1. Initial program 8.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    if 7.170478586447025e-264 < b < 3.771181145902542e+84

    1. Initial program 34.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+34.1

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    4. Simplified16.5

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
    5. Using strategy rm

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

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

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

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}{2 \cdot a}\]

    8. Applied times-frac16.5

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    9. Applied associate-/l*16.7

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}\]

    10. Simplified16.1

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]
    11. Using strategy rm

    12. Applied times-frac16.1

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\left(\frac{2}{4} \cdot \frac{a}{a \cdot c}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    13. Simplified9.7

      \[\leadsto \frac{\frac{1}{1}}{\left(\frac{2}{4} \cdot \color{blue}{\frac{1}{c}}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]
    14. Using strategy rm

    15. Applied div-inv9.7

      \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1}{\left(\frac{2}{4} \cdot \frac{1}{c}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}\]

    16. Simplified9.7

      \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\frac{1}{\frac{2}{4}}}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}}\]

    if 3.771181145902542e+84 < b

    1. Initial program 58.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 2.9

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.2389466313579672 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 7.17047858644702483 \cdot 10^{-264}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.7711811459025421 \cdot 10^{84}:\\ \;\;\;\;1 \cdot \frac{\frac{1}{\frac{2}{4}}}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))