Average Error: 33.8 → 6.6
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.847204280282031663920354805138023860461 \cdot 10^{48}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.441935351115962169406457345797212275167 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1.406682295301811137671309943547484787165 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{2}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.847204280282031663920354805138023860461 \cdot 10^{48}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -8.441935351115962169406457345797212275167 \cdot 10^{-251}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{elif}\;b \le 1.406682295301811137671309943547484787165 \cdot 10^{103}:\\
\;\;\;\;\frac{\frac{4 \cdot c}{2}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r76367 = b;
        double r76368 = -r76367;
        double r76369 = r76367 * r76367;
        double r76370 = 4.0;
        double r76371 = a;
        double r76372 = c;
        double r76373 = r76371 * r76372;
        double r76374 = r76370 * r76373;
        double r76375 = r76369 - r76374;
        double r76376 = sqrt(r76375);
        double r76377 = r76368 + r76376;
        double r76378 = 2.0;
        double r76379 = r76378 * r76371;
        double r76380 = r76377 / r76379;
        return r76380;
}
↓
double f(double a, double b, double c) {
        double r76381 = b;
        double r76382 = -2.8472042802820317e+48;
        bool r76383 = r76381 <= r76382;
        double r76384 = 1.0;
        double r76385 = c;
        double r76386 = r76385 / r76381;
        double r76387 = a;
        double r76388 = r76381 / r76387;
        double r76389 = r76386 - r76388;
        double r76390 = r76384 * r76389;
        double r76391 = -8.441935351115962e-251;
        bool r76392 = r76381 <= r76391;
        double r76393 = 1.0;
        double r76394 = 2.0;
        double r76395 = r76394 * r76387;
        double r76396 = r76381 * r76381;
        double r76397 = 4.0;
        double r76398 = r76387 * r76385;
        double r76399 = r76397 * r76398;
        double r76400 = r76396 - r76399;
        double r76401 = sqrt(r76400);
        double r76402 = r76401 - r76381;
        double r76403 = r76395 / r76402;
        double r76404 = r76393 / r76403;
        double r76405 = 1.4066822953018111e+103;
        bool r76406 = r76381 <= r76405;
        double r76407 = r76397 * r76385;
        double r76408 = r76407 / r76394;
        double r76409 = -r76381;
        double r76410 = r76409 - r76401;
        double r76411 = r76408 / r76410;
        double r76412 = -1.0;
        double r76413 = r76412 * r76386;
        double r76414 = r76406 ? r76411 : r76413;
        double r76415 = r76392 ? r76404 : r76414;
        double r76416 = r76383 ? r76390 : r76415;
        return r76416;
}

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

Original33.8
Target20.6
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.8472042802820317e+48

    1. Initial program 38.1

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

    2. Taylor expanded around -inf 5.2

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

    3. Simplified5.2

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

    if -2.8472042802820317e+48 < b < -8.441935351115962e-251

    1. Initial program 9.0

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

    3. Applied clear-num9.1

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

    4. Simplified9.1

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

    if -8.441935351115962e-251 < b < 1.4066822953018111e+103

    1. Initial program 30.1

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

    3. Applied flip-+30.2

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

    4. Simplified15.8

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

    6. Applied div-inv15.8

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

    8. Applied pow115.8

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

    9. Applied pow115.8

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

    10. Applied pow-prod-down15.8

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

    11. Simplified15.0

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

    13. Applied times-frac8.9

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

    14. Simplified8.9

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

    if 1.4066822953018111e+103 < b

    1. Initial program 59.1

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

    2. Taylor expanded around inf 2.2

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

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.847204280282031663920354805138023860461 \cdot 10^{48}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.441935351115962169406457345797212275167 \cdot 10^{-251}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{elif}\;b \le 1.406682295301811137671309943547484787165 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{4 \cdot c}{2}}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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)))