Average Error: 33.6 → 9.6
Time: 25.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.768773924260542 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.9878793504095505 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]
\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 -9.768773924260542 \cdot 10^{+151}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 5.9878793504095505 \cdot 10^{-84}:\\
\;\;\;\;\frac{\frac{\sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2191346 = b;
        double r2191347 = -r2191346;
        double r2191348 = r2191346 * r2191346;
        double r2191349 = 4.0;
        double r2191350 = a;
        double r2191351 = c;
        double r2191352 = r2191350 * r2191351;
        double r2191353 = r2191349 * r2191352;
        double r2191354 = r2191348 - r2191353;
        double r2191355 = sqrt(r2191354);
        double r2191356 = r2191347 + r2191355;
        double r2191357 = 2.0;
        double r2191358 = r2191357 * r2191350;
        double r2191359 = r2191356 / r2191358;
        return r2191359;
}
↓
double f(double a, double b, double c) {
        double r2191360 = b;
        double r2191361 = -9.768773924260542e+151;
        bool r2191362 = r2191360 <= r2191361;
        double r2191363 = c;
        double r2191364 = r2191363 / r2191360;
        double r2191365 = a;
        double r2191366 = r2191360 / r2191365;
        double r2191367 = r2191364 - r2191366;
        double r2191368 = 2.0;
        double r2191369 = r2191367 * r2191368;
        double r2191370 = r2191369 / r2191368;
        double r2191371 = 5.9878793504095505e-84;
        bool r2191372 = r2191360 <= r2191371;
        double r2191373 = r2191363 * r2191365;
        double r2191374 = -4.0;
        double r2191375 = r2191373 * r2191374;
        double r2191376 = r2191360 * r2191360;
        double r2191377 = r2191375 + r2191376;
        double r2191378 = sqrt(r2191377);
        double r2191379 = r2191378 - r2191360;
        double r2191380 = r2191379 / r2191365;
        double r2191381 = r2191380 / r2191368;
        double r2191382 = -2.0;
        double r2191383 = r2191364 * r2191382;
        double r2191384 = r2191383 / r2191368;
        double r2191385 = r2191372 ? r2191381 : r2191384;
        double r2191386 = r2191362 ? r2191370 : r2191385;
        return r2191386;
}

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

Original33.6
Target20.4
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.768773924260542e+151

    1. Initial program 60.2

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

    2. Simplified60.2

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

    4. Applied div-inv60.2

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

    5. Taylor expanded around -inf 1.7

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

    6. Simplified1.7

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

    if -9.768773924260542e+151 < b < 5.9878793504095505e-84

    1. Initial program 11.8

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

    2. Simplified11.8

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

    4. Applied div-inv11.9

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

    6. Applied associate-*r/11.8

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

    7. Simplified11.8

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

    if 5.9878793504095505e-84 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 9.3

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.768773924260542 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 5.9878793504095505 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))