Average Error: 34.1 → 10.6
Time: 5.0s
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 r72272 = b;
        double r72273 = -r72272;
        double r72274 = r72272 * r72272;
        double r72275 = 4.0;
        double r72276 = a;
        double r72277 = c;
        double r72278 = r72276 * r72277;
        double r72279 = r72275 * r72278;
        double r72280 = r72274 - r72279;
        double r72281 = sqrt(r72280);
        double r72282 = r72273 + r72281;
        double r72283 = 2.0;
        double r72284 = r72283 * r72276;
        double r72285 = r72282 / r72284;
        return r72285;
}
↓
double f(double a, double b, double c) {
        double r72286 = b;
        double r72287 = -2.237522594933402e+57;
        bool r72288 = r72286 <= r72287;
        double r72289 = 1.0;
        double r72290 = c;
        double r72291 = r72290 / r72286;
        double r72292 = a;
        double r72293 = r72286 / r72292;
        double r72294 = r72291 - r72293;
        double r72295 = r72289 * r72294;
        double r72296 = 8.679707852111266e-40;
        bool r72297 = r72286 <= r72296;
        double r72298 = -r72286;
        double r72299 = r72286 * r72286;
        double r72300 = 4.0;
        double r72301 = r72292 * r72290;
        double r72302 = r72300 * r72301;
        double r72303 = r72299 - r72302;
        double r72304 = sqrt(r72303);
        double r72305 = r72298 + r72304;
        double r72306 = 2.0;
        double r72307 = r72306 * r72292;
        double r72308 = r72305 / r72307;
        double r72309 = -1.0;
        double r72310 = r72309 * r72291;
        double r72311 = r72297 ? r72308 : r72310;
        double r72312 = r72288 ? r72295 : r72311;
        return r72312;
}

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)))