Average Error: 34.6 → 10.7
Time: 28.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\
\;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6174261 = b;
        double r6174262 = -r6174261;
        double r6174263 = r6174261 * r6174261;
        double r6174264 = 4.0;
        double r6174265 = a;
        double r6174266 = r6174264 * r6174265;
        double r6174267 = c;
        double r6174268 = r6174266 * r6174267;
        double r6174269 = r6174263 - r6174268;
        double r6174270 = sqrt(r6174269);
        double r6174271 = r6174262 + r6174270;
        double r6174272 = 2.0;
        double r6174273 = r6174272 * r6174265;
        double r6174274 = r6174271 / r6174273;
        return r6174274;
}
↓
double f(double a, double b, double c) {
        double r6174275 = b;
        double r6174276 = -2.7668189408748547e+100;
        bool r6174277 = r6174275 <= r6174276;
        double r6174278 = c;
        double r6174279 = r6174278 / r6174275;
        double r6174280 = a;
        double r6174281 = r6174275 / r6174280;
        double r6174282 = r6174279 - r6174281;
        double r6174283 = 1.0;
        double r6174284 = r6174282 * r6174283;
        double r6174285 = 7.923524897992037e-153;
        bool r6174286 = r6174275 <= r6174285;
        double r6174287 = 1.0;
        double r6174288 = 2.0;
        double r6174289 = r6174280 * r6174288;
        double r6174290 = r6174287 / r6174289;
        double r6174291 = r6174275 * r6174275;
        double r6174292 = r6174280 * r6174278;
        double r6174293 = 4.0;
        double r6174294 = r6174292 * r6174293;
        double r6174295 = r6174291 - r6174294;
        double r6174296 = sqrt(r6174295);
        double r6174297 = r6174296 - r6174275;
        double r6174298 = r6174287 / r6174297;
        double r6174299 = r6174290 / r6174298;
        double r6174300 = -1.0;
        double r6174301 = r6174279 * r6174300;
        double r6174302 = r6174286 ? r6174299 : r6174301;
        double r6174303 = r6174277 ? r6174284 : r6174302;
        return r6174303;
}

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

Original34.6
Target21.1
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.7668189408748547e+100

    1. Initial program 47.2

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

    2. Simplified47.2

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

    3. Taylor expanded around -inf 4.0

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

    4. Simplified4.0

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

    if -2.7668189408748547e+100 < b < 7.923524897992037e-153

    1. Initial program 10.8

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

    2. Simplified10.9

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

    4. Applied clear-num11.0

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}\]
    5. Using strategy rm

    6. Applied div-inv11.1

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

    7. Applied associate-/r*11.0

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

    if 7.923524897992037e-153 < b

    1. Initial program 50.5

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

    2. Simplified50.5

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

    3. Taylor expanded around inf 12.7

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.766818940874854722177248139872145176232 \cdot 10^{100}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 7.923524897992036987166355557663274472861 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{1}{a \cdot 2}}{\frac{1}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))