Average Error: 34.2 → 6.5
Time: 12.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.193871707188482833811342019428468815697 \cdot 10^{-287}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.521192511657275894218075856706322414394 \cdot 10^{99}:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -4.193871707188482833811342019428468815697 \cdot 10^{-287}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 2.521192511657275894218075856706322414394 \cdot 10^{99}:\\
\;\;\;\;{\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\\

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

\end{array}
double f(double a, double b, double c) {
        double r126288 = b;
        double r126289 = -r126288;
        double r126290 = r126288 * r126288;
        double r126291 = 4.0;
        double r126292 = a;
        double r126293 = r126291 * r126292;
        double r126294 = c;
        double r126295 = r126293 * r126294;
        double r126296 = r126290 - r126295;
        double r126297 = sqrt(r126296);
        double r126298 = r126289 + r126297;
        double r126299 = 2.0;
        double r126300 = r126299 * r126292;
        double r126301 = r126298 / r126300;
        return r126301;
}
↓
double f(double a, double b, double c) {
        double r126302 = b;
        double r126303 = -1.8440038131758226e+119;
        bool r126304 = r126302 <= r126303;
        double r126305 = 1.0;
        double r126306 = c;
        double r126307 = r126306 / r126302;
        double r126308 = a;
        double r126309 = r126302 / r126308;
        double r126310 = r126307 - r126309;
        double r126311 = r126305 * r126310;
        double r126312 = -4.193871707188483e-287;
        bool r126313 = r126302 <= r126312;
        double r126314 = -r126302;
        double r126315 = r126302 * r126302;
        double r126316 = 4.0;
        double r126317 = r126316 * r126308;
        double r126318 = r126317 * r126306;
        double r126319 = r126315 - r126318;
        double r126320 = sqrt(r126319);
        double r126321 = r126314 + r126320;
        double r126322 = 1.0;
        double r126323 = 2.0;
        double r126324 = r126323 * r126308;
        double r126325 = r126322 / r126324;
        double r126326 = r126321 * r126325;
        double r126327 = 2.521192511657276e+99;
        bool r126328 = r126302 <= r126327;
        double r126329 = r126323 * r126306;
        double r126330 = r126314 - r126320;
        double r126331 = r126329 / r126330;
        double r126332 = pow(r126331, r126322);
        double r126333 = -1.0;
        double r126334 = r126333 * r126307;
        double r126335 = r126328 ? r126332 : r126334;
        double r126336 = r126313 ? r126326 : r126335;
        double r126337 = r126304 ? r126311 : r126336;
        return r126337;
}

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

Original34.2
Target21.1
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8440038131758226e+119

    1. Initial program 51.6

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

    2. Taylor expanded around -inf 3.0

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

    3. Simplified3.0

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

    if -1.8440038131758226e+119 < b < -4.193871707188483e-287

    1. Initial program 8.2

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

    3. Applied div-inv8.4

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

    if -4.193871707188483e-287 < b < 2.521192511657276e+99

    1. Initial program 31.6

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

    3. Applied flip-+31.7

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

    4. Simplified16.7

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

    6. Applied div-inv16.8

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

    8. Applied pow116.8

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

    9. Applied pow116.8

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

    10. Applied pow-prod-down16.8

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

    11. Simplified16.0

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

    12. Taylor expanded around 0 9.4

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

    if 2.521192511657276e+99 < b

    1. Initial program 59.4

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

    2. Taylor expanded around inf 2.6

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.844003813175822562359270713493973222617 \cdot 10^{119}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.193871707188482833811342019428468815697 \cdot 10^{-287}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 2.521192511657275894218075856706322414394 \cdot 10^{99}:\\ \;\;\;\;{\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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)))