Average Error: 33.7 → 10.2
Time: 20.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.3248915655872564 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.3248915655872564 \cdot 10^{+79}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4127239 = b;
        double r4127240 = -r4127239;
        double r4127241 = r4127239 * r4127239;
        double r4127242 = 4.0;
        double r4127243 = a;
        double r4127244 = r4127242 * r4127243;
        double r4127245 = c;
        double r4127246 = r4127244 * r4127245;
        double r4127247 = r4127241 - r4127246;
        double r4127248 = sqrt(r4127247);
        double r4127249 = r4127240 + r4127248;
        double r4127250 = 2.0;
        double r4127251 = r4127250 * r4127243;
        double r4127252 = r4127249 / r4127251;
        return r4127252;
}
↓
double f(double a, double b, double c) {
        double r4127253 = b;
        double r4127254 = -5.3248915655872564e+79;
        bool r4127255 = r4127253 <= r4127254;
        double r4127256 = c;
        double r4127257 = r4127256 / r4127253;
        double r4127258 = a;
        double r4127259 = r4127253 / r4127258;
        double r4127260 = r4127257 - r4127259;
        double r4127261 = 4.2796532586596585e-91;
        bool r4127262 = r4127253 <= r4127261;
        double r4127263 = -r4127253;
        double r4127264 = r4127253 * r4127253;
        double r4127265 = 4.0;
        double r4127266 = r4127265 * r4127258;
        double r4127267 = r4127256 * r4127266;
        double r4127268 = r4127264 - r4127267;
        double r4127269 = sqrt(r4127268);
        double r4127270 = r4127263 + r4127269;
        double r4127271 = 0.5;
        double r4127272 = r4127271 / r4127258;
        double r4127273 = r4127270 * r4127272;
        double r4127274 = -r4127257;
        double r4127275 = r4127262 ? r4127273 : r4127274;
        double r4127276 = r4127255 ? r4127260 : r4127275;
        return r4127276;
}

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

Original33.7
Target20.5
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.3248915655872564e+79

    1. Initial program 41.1

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

    2. Taylor expanded around -inf 4.6

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

    if -5.3248915655872564e+79 < b < 4.2796532586596585e-91

    1. Initial program 13.0

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

    3. Applied div-inv13.1

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

    4. Simplified13.1

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

    if 4.2796532586596585e-91 < b

    1. Initial program 52.0

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

    3. Applied div-inv52.0

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

    4. Simplified52.0

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

    5. Taylor expanded around inf 9.6

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

    6. Simplified9.6

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.3248915655872564 \cdot 10^{+79}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.2796532586596585 \cdot 10^{-91}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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