Average Error: 34.2 → 6.7
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\left(\frac{2}{4} \cdot \frac{1}{c}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\
\;\;\;\;\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)}\\

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

\end{array}
double f(double a, double b, double c) {
        double r96265 = b;
        double r96266 = -r96265;
        double r96267 = r96265 * r96265;
        double r96268 = 4.0;
        double r96269 = a;
        double r96270 = c;
        double r96271 = r96269 * r96270;
        double r96272 = r96268 * r96271;
        double r96273 = r96267 - r96272;
        double r96274 = sqrt(r96273);
        double r96275 = r96266 + r96274;
        double r96276 = 2.0;
        double r96277 = r96276 * r96269;
        double r96278 = r96275 / r96277;
        return r96278;
}
↓
double f(double a, double b, double c) {
        double r96279 = b;
        double r96280 = -4.739386840053889e+131;
        bool r96281 = r96279 <= r96280;
        double r96282 = 1.0;
        double r96283 = c;
        double r96284 = r96283 / r96279;
        double r96285 = a;
        double r96286 = r96279 / r96285;
        double r96287 = r96284 - r96286;
        double r96288 = r96282 * r96287;
        double r96289 = -2.1023086245622604e-293;
        bool r96290 = r96279 <= r96289;
        double r96291 = -r96279;
        double r96292 = r96279 * r96279;
        double r96293 = 4.0;
        double r96294 = r96285 * r96283;
        double r96295 = r96293 * r96294;
        double r96296 = r96292 - r96295;
        double r96297 = sqrt(r96296);
        double r96298 = r96291 + r96297;
        double r96299 = 1.0;
        double r96300 = 2.0;
        double r96301 = r96300 * r96285;
        double r96302 = r96299 / r96301;
        double r96303 = r96298 * r96302;
        double r96304 = 6.09240124692818e+90;
        bool r96305 = r96279 <= r96304;
        double r96306 = r96300 / r96293;
        double r96307 = r96299 / r96283;
        double r96308 = r96306 * r96307;
        double r96309 = r96291 - r96297;
        double r96310 = r96308 * r96309;
        double r96311 = r96299 / r96310;
        double r96312 = -1.0;
        double r96313 = r96312 * r96284;
        double r96314 = r96305 ? r96311 : r96313;
        double r96315 = r96290 ? r96303 : r96314;
        double r96316 = r96281 ? r96288 : r96315;
        return r96316;
}

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
Target21.2
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -4.739386840053889e+131

    1. Initial program 55.7

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

    2. Taylor expanded around -inf 2.4

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

    3. Simplified2.4

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

    if -4.739386840053889e+131 < b < -2.1023086245622604e-293

    1. Initial program 9.2

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

    3. Applied div-inv9.4

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

    if -2.1023086245622604e-293 < b < 6.09240124692818e+90

    1. Initial program 31.3

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

    3. Applied flip-+31.3

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

    4. Simplified16.0

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

    6. Applied *-un-lft-identity16.0

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

    7. Applied *-un-lft-identity16.0

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

    8. Applied times-frac16.0

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

    9. Applied associate-/l*16.2

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

    10. Simplified15.6

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

    12. Applied times-frac15.6

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

    13. Simplified8.8

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

    if 6.09240124692818e+90 < b

    1. Initial program 59.2

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

    2. Taylor expanded around inf 3.0

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

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.739386840053888999010128333992752158317 \cdot 10^{131}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.102308624562260429751103075089775725609 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.092401246928180338651406165764155275885 \cdot 10^{90}:\\ \;\;\;\;\frac{1}{\left(\frac{2}{4} \cdot \frac{1}{c}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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)))