Average Error: 34.2 → 15.1
Time: 12.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.3358786167585806 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\ \mathbf{elif}\;b \le 1.94263717460376656 \cdot 10^{24}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{2}}{a}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.3358786167585806 \cdot 10^{154}:\\
\;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\

\mathbf{elif}\;b \le 1.94263717460376656 \cdot 10^{24}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r96243 = b;
        double r96244 = -r96243;
        double r96245 = r96243 * r96243;
        double r96246 = 4.0;
        double r96247 = a;
        double r96248 = r96246 * r96247;
        double r96249 = c;
        double r96250 = r96248 * r96249;
        double r96251 = r96245 - r96250;
        double r96252 = sqrt(r96251);
        double r96253 = r96244 + r96252;
        double r96254 = 2.0;
        double r96255 = r96254 * r96247;
        double r96256 = r96253 / r96255;
        return r96256;
}
↓
double f(double a, double b, double c) {
        double r96257 = b;
        double r96258 = -1.3358786167585806e+154;
        bool r96259 = r96257 <= r96258;
        double r96260 = 2.0;
        double r96261 = a;
        double r96262 = c;
        double r96263 = r96261 * r96262;
        double r96264 = r96263 / r96257;
        double r96265 = r96260 * r96264;
        double r96266 = 2.0;
        double r96267 = r96266 * r96257;
        double r96268 = r96265 - r96267;
        double r96269 = r96268 / r96260;
        double r96270 = r96269 / r96261;
        double r96271 = 1.9426371746037666e+24;
        bool r96272 = r96257 <= r96271;
        double r96273 = r96257 * r96257;
        double r96274 = 4.0;
        double r96275 = r96274 * r96261;
        double r96276 = r96275 * r96262;
        double r96277 = r96273 - r96276;
        double r96278 = sqrt(r96277);
        double r96279 = r96278 - r96257;
        double r96280 = r96279 / r96260;
        double r96281 = r96280 / r96261;
        double r96282 = -2.0;
        double r96283 = r96282 * r96264;
        double r96284 = r96283 / r96260;
        double r96285 = r96284 / r96261;
        double r96286 = r96272 ? r96281 : r96285;
        double r96287 = r96259 ? r96270 : r96286;
        return r96287;
}

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
Target21.2
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3358786167585806e+154

    1. Initial program 64.0

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

    2. Simplified64.0

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

    3. Taylor expanded around -inf 9.8

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

    if -1.3358786167585806e+154 < b < 1.9426371746037666e+24

    1. Initial program 16.1

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

    2. Simplified16.1

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

    if 1.9426371746037666e+24 < b

    1. Initial program 56.4

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

    2. Simplified56.4

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

    3. Taylor expanded around inf 15.2

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

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3358786167585806 \cdot 10^{154}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}{2}}{a}\\ \mathbf{elif}\;b \le 1.94263717460376656 \cdot 10^{24}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot c}{b}}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))