Average Error: 34.2 → 10.5
Time: 4.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r91306 = b;
        double r91307 = -r91306;
        double r91308 = r91306 * r91306;
        double r91309 = 4.0;
        double r91310 = a;
        double r91311 = c;
        double r91312 = r91310 * r91311;
        double r91313 = r91309 * r91312;
        double r91314 = r91308 - r91313;
        double r91315 = sqrt(r91314);
        double r91316 = r91307 - r91315;
        double r91317 = 2.0;
        double r91318 = r91317 * r91310;
        double r91319 = r91316 / r91318;
        return r91319;
}
↓
double f(double a, double b, double c) {
        double r91320 = b;
        double r91321 = -2.763265898390889e-72;
        bool r91322 = r91320 <= r91321;
        double r91323 = -1.0;
        double r91324 = c;
        double r91325 = r91324 / r91320;
        double r91326 = r91323 * r91325;
        double r91327 = 6.568668442325333e+48;
        bool r91328 = r91320 <= r91327;
        double r91329 = -r91320;
        double r91330 = r91320 * r91320;
        double r91331 = 4.0;
        double r91332 = a;
        double r91333 = r91332 * r91324;
        double r91334 = r91331 * r91333;
        double r91335 = r91330 - r91334;
        double r91336 = sqrt(r91335);
        double r91337 = r91329 - r91336;
        double r91338 = 1.0;
        double r91339 = 2.0;
        double r91340 = r91339 * r91332;
        double r91341 = r91338 / r91340;
        double r91342 = r91337 * r91341;
        double r91343 = 1.0;
        double r91344 = r91320 / r91332;
        double r91345 = r91325 - r91344;
        double r91346 = r91343 * r91345;
        double r91347 = r91328 ? r91342 : r91346;
        double r91348 = r91322 ? r91326 : r91347;
        return r91348;
}

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
Target21.3
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.763265898390889e-72

    1. Initial program 52.9

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

    2. Taylor expanded around -inf 9.2

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

    if -2.763265898390889e-72 < b < 6.568668442325333e+48

    1. Initial program 14.1

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

    3. Applied div-inv14.2

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

    if 6.568668442325333e+48 < b

    1. Initial program 39.1

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

    2. Taylor expanded around inf 5.4

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

    3. Simplified5.4

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.763265898390888870746272508253147533908 \cdot 10^{-72}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.568668442325333133869311590844159217228 \cdot 10^{48}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))