Average Error: 33.8 → 10.1
Time: 8.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.12428337420519208 \cdot 10^{57}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r88269 = b;
        double r88270 = -r88269;
        double r88271 = r88269 * r88269;
        double r88272 = 4.0;
        double r88273 = a;
        double r88274 = c;
        double r88275 = r88273 * r88274;
        double r88276 = r88272 * r88275;
        double r88277 = r88271 - r88276;
        double r88278 = sqrt(r88277);
        double r88279 = r88270 + r88278;
        double r88280 = 2.0;
        double r88281 = r88280 * r88273;
        double r88282 = r88279 / r88281;
        return r88282;
}
↓
double f(double a, double b, double c) {
        double r88283 = b;
        double r88284 = -3.124283374205192e+57;
        bool r88285 = r88283 <= r88284;
        double r88286 = 1.0;
        double r88287 = c;
        double r88288 = r88287 / r88283;
        double r88289 = a;
        double r88290 = r88283 / r88289;
        double r88291 = r88288 - r88290;
        double r88292 = r88286 * r88291;
        double r88293 = 3.84613441880261e-81;
        bool r88294 = r88283 <= r88293;
        double r88295 = 2.0;
        double r88296 = pow(r88283, r88295);
        double r88297 = 4.0;
        double r88298 = r88289 * r88287;
        double r88299 = r88297 * r88298;
        double r88300 = r88296 - r88299;
        double r88301 = sqrt(r88300);
        double r88302 = r88301 - r88283;
        double r88303 = 2.0;
        double r88304 = r88302 / r88303;
        double r88305 = r88304 / r88289;
        double r88306 = -1.0;
        double r88307 = r88306 * r88288;
        double r88308 = r88294 ? r88305 : r88307;
        double r88309 = r88285 ? r88292 : r88308;
        return r88309;
}

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

Original33.8
Target20.4
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.124283374205192e+57

    1. Initial program 39.5

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

    2. Simplified39.5

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

    3. Taylor expanded around -inf 5.4

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

    4. Simplified5.4

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

    if -3.124283374205192e+57 < b < 3.84613441880261e-81

    1. Initial program 12.7

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

    2. Simplified12.7

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

    3. Taylor expanded around 0 12.7

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

    if 3.84613441880261e-81 < b

    1. Initial program 53.0

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

    2. Simplified53.0

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

    3. Taylor expanded around inf 9.5

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 3.84613441880260993 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))