Average Error: 33.4 → 9.8
Time: 23.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.852138444177435 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{\left(-4 \cdot c\right) \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.852138444177435 \cdot 10^{-54}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{\left(-4 \cdot c\right) \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3005350 = b;
        double r3005351 = -r3005350;
        double r3005352 = r3005350 * r3005350;
        double r3005353 = 4.0;
        double r3005354 = a;
        double r3005355 = c;
        double r3005356 = r3005354 * r3005355;
        double r3005357 = r3005353 * r3005356;
        double r3005358 = r3005352 - r3005357;
        double r3005359 = sqrt(r3005358);
        double r3005360 = r3005351 - r3005359;
        double r3005361 = 2.0;
        double r3005362 = r3005361 * r3005354;
        double r3005363 = r3005360 / r3005362;
        return r3005363;
}
↓
double f(double a, double b, double c) {
        double r3005364 = b;
        double r3005365 = -2.852138444177435e-54;
        bool r3005366 = r3005364 <= r3005365;
        double r3005367 = c;
        double r3005368 = r3005367 / r3005364;
        double r3005369 = -r3005368;
        double r3005370 = 6.359263193477048e+137;
        bool r3005371 = r3005364 <= r3005370;
        double r3005372 = -r3005364;
        double r3005373 = -4.0;
        double r3005374 = r3005373 * r3005367;
        double r3005375 = a;
        double r3005376 = r3005374 * r3005375;
        double r3005377 = r3005364 * r3005364;
        double r3005378 = r3005376 + r3005377;
        double r3005379 = sqrt(r3005378);
        double r3005380 = r3005372 - r3005379;
        double r3005381 = 0.5;
        double r3005382 = r3005381 / r3005375;
        double r3005383 = r3005380 * r3005382;
        double r3005384 = r3005364 / r3005375;
        double r3005385 = r3005368 - r3005384;
        double r3005386 = r3005371 ? r3005383 : r3005385;
        double r3005387 = r3005366 ? r3005369 : r3005386;
        return r3005387;
}

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

Original33.4
Target20.8
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.852138444177435e-54

    1. Initial program 53.4

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

    3. Applied sub-neg53.4

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

    4. Simplified53.4

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

    6. Applied div-inv53.4

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

    7. Simplified53.4

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

    8. Taylor expanded around -inf 8.3

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

    9. Simplified8.3

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

    if -2.852138444177435e-54 < b < 6.359263193477048e+137

    1. Initial program 12.6

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

    3. Applied sub-neg12.6

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

    4. Simplified12.7

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

    6. Applied div-inv12.8

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

    7. Simplified12.8

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

    if 6.359263193477048e+137 < 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. Using strategy rm

    3. Applied sub-neg53.0

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

    4. Simplified53.1

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

    5. Taylor expanded around inf 2.5

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.852138444177435 \cdot 10^{-54}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 6.359263193477048 \cdot 10^{+137}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{\left(-4 \cdot c\right) \cdot a + b \cdot b}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 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)))