Average Error: 34.5 → 9.7
Time: 47.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1.0\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0\\

\mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1.0\\

\end{array}
double f(double a, double b, double c) {
        double r4159214 = b;
        double r4159215 = -r4159214;
        double r4159216 = r4159214 * r4159214;
        double r4159217 = 4.0;
        double r4159218 = a;
        double r4159219 = c;
        double r4159220 = r4159218 * r4159219;
        double r4159221 = r4159217 * r4159220;
        double r4159222 = r4159216 - r4159221;
        double r4159223 = sqrt(r4159222);
        double r4159224 = r4159215 + r4159223;
        double r4159225 = 2.0;
        double r4159226 = r4159225 * r4159218;
        double r4159227 = r4159224 / r4159226;
        return r4159227;
}
↓
double f(double a, double b, double c) {
        double r4159228 = b;
        double r4159229 = -5.517926393801403e+142;
        bool r4159230 = r4159228 <= r4159229;
        double r4159231 = c;
        double r4159232 = r4159231 / r4159228;
        double r4159233 = a;
        double r4159234 = r4159228 / r4159233;
        double r4159235 = r4159232 - r4159234;
        double r4159236 = 1.0;
        double r4159237 = r4159235 * r4159236;
        double r4159238 = 1.3635892865650846e-93;
        bool r4159239 = r4159228 <= r4159238;
        double r4159240 = r4159228 * r4159228;
        double r4159241 = r4159233 * r4159231;
        double r4159242 = 4.0;
        double r4159243 = r4159241 * r4159242;
        double r4159244 = r4159240 - r4159243;
        double r4159245 = sqrt(r4159244);
        double r4159246 = -r4159228;
        double r4159247 = r4159245 + r4159246;
        double r4159248 = 1.0;
        double r4159249 = 2.0;
        double r4159250 = r4159233 * r4159249;
        double r4159251 = r4159248 / r4159250;
        double r4159252 = r4159247 * r4159251;
        double r4159253 = -1.0;
        double r4159254 = r4159232 * r4159253;
        double r4159255 = r4159239 ? r4159252 : r4159254;
        double r4159256 = r4159230 ? r4159237 : r4159255;
        return r4159256;
}

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

Original34.5
Target20.9
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.517926393801403e+142

    1. Initial program 59.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+63.9

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}}}{2.0 \cdot a}\]

    4. Simplified63.9

      \[\leadsto \frac{\frac{\color{blue}{\left(b \cdot b - b \cdot b\right) + 4.0 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}}{2.0 \cdot a}\]

    5. Taylor expanded around -inf 2.7

      \[\leadsto \color{blue}{1.0 \cdot \frac{c}{b} - 1.0 \cdot \frac{b}{a}}\]

    6. Simplified2.7

      \[\leadsto \color{blue}{1.0 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -5.517926393801403e+142 < b < 1.3635892865650846e-93

    1. Initial program 11.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv11.9

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2.0 \cdot a}}\]

    if 1.3635892865650846e-93 < b

    1. Initial program 52.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]

    2. Taylor expanded around inf 9.1

      \[\leadsto \color{blue}{-1.0 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.517926393801403 \cdot 10^{+142}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1.0\\ \mathbf{elif}\;b \le 1.3635892865650846 \cdot 10^{-93}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1.0\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))