Average Error: 34.3 → 9.8
Time: 15.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.860817974822627954816964027859042173562 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2000198799923726.5:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.860817974822627954816964027859042173562 \cdot 10^{-201}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2000198799923726.5:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r70275 = b;
        double r70276 = -r70275;
        double r70277 = r70275 * r70275;
        double r70278 = 4.0;
        double r70279 = a;
        double r70280 = c;
        double r70281 = r70279 * r70280;
        double r70282 = r70278 * r70281;
        double r70283 = r70277 - r70282;
        double r70284 = sqrt(r70283);
        double r70285 = r70276 - r70284;
        double r70286 = 2.0;
        double r70287 = r70286 * r70279;
        double r70288 = r70285 / r70287;
        return r70288;
}
↓
double f(double a, double b, double c) {
        double r70289 = b;
        double r70290 = -7.359940312872037e+54;
        bool r70291 = r70289 <= r70290;
        double r70292 = -1.0;
        double r70293 = c;
        double r70294 = r70293 / r70289;
        double r70295 = r70292 * r70294;
        double r70296 = -2.860817974822628e-201;
        bool r70297 = r70289 <= r70296;
        double r70298 = 4.0;
        double r70299 = a;
        double r70300 = r70299 * r70293;
        double r70301 = r70298 * r70300;
        double r70302 = r70289 * r70289;
        double r70303 = r70302 - r70301;
        double r70304 = sqrt(r70303);
        double r70305 = r70304 - r70289;
        double r70306 = r70301 / r70305;
        double r70307 = 2.0;
        double r70308 = r70307 * r70299;
        double r70309 = r70306 / r70308;
        double r70310 = 2000198799923726.5;
        bool r70311 = r70289 <= r70310;
        double r70312 = -r70289;
        double r70313 = r70312 - r70304;
        double r70314 = 1.0;
        double r70315 = r70314 / r70308;
        double r70316 = r70313 * r70315;
        double r70317 = 1.0;
        double r70318 = r70289 / r70299;
        double r70319 = r70294 - r70318;
        double r70320 = r70317 * r70319;
        double r70321 = r70311 ? r70316 : r70320;
        double r70322 = r70297 ? r70309 : r70321;
        double r70323 = r70291 ? r70295 : r70322;
        return r70323;
}

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

Original34.3
Target20.9
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.359940312872037e+54

    1. Initial program 57.6

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

    2. Taylor expanded around -inf 3.8

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

    if -7.359940312872037e+54 < b < -2.860817974822628e-201

    1. Initial program 36.5

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

    3. Applied flip--36.5

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

    4. Simplified18.6

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified18.6

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity18.6

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b\right)}}}{2 \cdot a}\]

    if -2.860817974822628e-201 < b < 2000198799923726.5

    1. Initial program 11.6

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

    3. Applied div-inv11.7

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

    if 2000198799923726.5 < b

    1. Initial program 34.0

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

    2. Taylor expanded around inf 6.8

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

    3. Simplified6.8

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.359940312872037386934109274309219747139 \cdot 10^{54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.860817974822627954816964027859042173562 \cdot 10^{-201}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2000198799923726.5:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))