Average Error: 34.1 → 10.6
Time: 9.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2375225949334019 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.67970785211126629 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.2375225949334019 \cdot 10^{57}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 8.67970785211126629 \cdot 10^{-40}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r342 = b;
        double r343 = -r342;
        double r344 = r342 * r342;
        double r345 = 4.0;
        double r346 = a;
        double r347 = c;
        double r348 = r346 * r347;
        double r349 = r345 * r348;
        double r350 = r344 - r349;
        double r351 = sqrt(r350);
        double r352 = r343 + r351;
        double r353 = 2.0;
        double r354 = r353 * r346;
        double r355 = r352 / r354;
        return r355;
}
↓
double f(double a, double b, double c) {
        double r356 = b;
        double r357 = -2.237522594933402e+57;
        bool r358 = r356 <= r357;
        double r359 = 1.0;
        double r360 = c;
        double r361 = r360 / r356;
        double r362 = a;
        double r363 = r356 / r362;
        double r364 = r361 - r363;
        double r365 = r359 * r364;
        double r366 = 8.679707852111266e-40;
        bool r367 = r356 <= r366;
        double r368 = -r356;
        double r369 = r356 * r356;
        double r370 = 4.0;
        double r371 = r362 * r360;
        double r372 = r370 * r371;
        double r373 = r369 - r372;
        double r374 = sqrt(r373);
        double r375 = r368 + r374;
        double r376 = 2.0;
        double r377 = r376 * r362;
        double r378 = r375 / r377;
        double r379 = -1.0;
        double r380 = r379 * r361;
        double r381 = r367 ? r378 : r380;
        double r382 = r358 ? r365 : r381;
        return r382;
}

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

Original34.1
Target20.9
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.237522594933402e+57

    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 5.5

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

    3. Simplified5.5

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

    if -2.237522594933402e+57 < b < 8.679707852111266e-40

    1. Initial program 15.3

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

    if 8.679707852111266e-40 < b

    1. Initial program 55.1

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2375225949334019 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.67970785211126629 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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)))