Average Error: 33.5 → 8.7
Time: 22.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.634052056998666 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 1.9081232567651569 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot -4\right)}{a}}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.3725796156555912 \cdot 10^{+127}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 4.634052056998666 \cdot 10^{-182}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{elif}\;b \le 1.9081232567651569 \cdot 10^{+149}:\\
\;\;\;\;\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot -4\right)}{a}}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} + b}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1991356 = b;
        double r1991357 = -r1991356;
        double r1991358 = r1991356 * r1991356;
        double r1991359 = 4.0;
        double r1991360 = a;
        double r1991361 = c;
        double r1991362 = r1991360 * r1991361;
        double r1991363 = r1991359 * r1991362;
        double r1991364 = r1991358 - r1991363;
        double r1991365 = sqrt(r1991364);
        double r1991366 = r1991357 + r1991365;
        double r1991367 = 2.0;
        double r1991368 = r1991367 * r1991360;
        double r1991369 = r1991366 / r1991368;
        return r1991369;
}
↓
double f(double a, double b, double c) {
        double r1991370 = b;
        double r1991371 = -1.3725796156555912e+127;
        bool r1991372 = r1991370 <= r1991371;
        double r1991373 = c;
        double r1991374 = r1991373 / r1991370;
        double r1991375 = a;
        double r1991376 = r1991370 / r1991375;
        double r1991377 = r1991374 - r1991376;
        double r1991378 = 2.0;
        double r1991379 = r1991377 * r1991378;
        double r1991380 = r1991379 / r1991378;
        double r1991381 = 4.634052056998666e-182;
        bool r1991382 = r1991370 <= r1991381;
        double r1991383 = r1991370 * r1991370;
        double r1991384 = 4.0;
        double r1991385 = r1991384 * r1991375;
        double r1991386 = r1991373 * r1991385;
        double r1991387 = r1991383 - r1991386;
        double r1991388 = sqrt(r1991387);
        double r1991389 = r1991388 / r1991375;
        double r1991390 = r1991389 - r1991376;
        double r1991391 = r1991390 / r1991378;
        double r1991392 = 1.9081232567651569e+149;
        bool r1991393 = r1991370 <= r1991392;
        double r1991394 = r1991383 - r1991383;
        double r1991395 = -4.0;
        double r1991396 = r1991375 * r1991395;
        double r1991397 = r1991373 * r1991396;
        double r1991398 = r1991394 + r1991397;
        double r1991399 = r1991398 / r1991375;
        double r1991400 = r1991397 + r1991383;
        double r1991401 = sqrt(r1991400);
        double r1991402 = r1991401 + r1991370;
        double r1991403 = r1991399 / r1991402;
        double r1991404 = r1991403 / r1991378;
        double r1991405 = -2.0;
        double r1991406 = r1991405 * r1991374;
        double r1991407 = r1991406 / r1991378;
        double r1991408 = r1991393 ? r1991404 : r1991407;
        double r1991409 = r1991382 ? r1991391 : r1991408;
        double r1991410 = r1991372 ? r1991380 : r1991409;
        return r1991410;
}

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

Original33.5
Target21.2
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.3725796156555912e+127

    1. Initial program 51.4

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

    2. Simplified51.4

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

    3. Taylor expanded around -inf 2.3

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

    4. Simplified2.3

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

    if -1.3725796156555912e+127 < b < 4.634052056998666e-182

    1. Initial program 10.6

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

    2. Simplified10.6

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

    4. Applied div-sub10.6

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

    if 4.634052056998666e-182 < b < 1.9081232567651569e+149

    1. Initial program 38.8

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

    2. Simplified38.8

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

    4. Applied *-un-lft-identity38.8

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

    5. Applied *-un-lft-identity38.8

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

    6. Applied distribute-lft-out--38.8

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

    7. Applied associate-/l*38.9

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

    9. Applied div-inv38.9

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

    10. Applied add-cube-cbrt38.9

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{a \cdot \frac{1}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}{2}\]

    11. Applied times-frac38.9

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}}{2}\]

    12. Simplified38.9

      \[\leadsto \frac{\color{blue}{\frac{1}{a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}}{2}\]

    13. Simplified38.9

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

    15. Applied flip--39.0

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

    16. Applied associate-*r/39.0

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

    17. Simplified13.5

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

    if 1.9081232567651569e+149 < b

    1. Initial program 62.2

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

    2. Simplified62.2

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

    4. Applied div-sub62.4

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

    5. Taylor expanded around inf 1.7

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 4.634052056998666 \cdot 10^{-182}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 1.9081232567651569 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot -4\right)}{a}}{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} + b}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))