Average Error: 34.3 → 9.1
Time: 6.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.568201128637223695690583924646661116118 \cdot 10^{55}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.546128927832045171835550197132658970541 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2 \cdot a}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{elif}\;b \le 9.748353205521284417498320088314179968082 \cdot 10^{77}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.568201128637223695690583924646661116118 \cdot 10^{55}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -4.546128927832045171835550197132658970541 \cdot 10^{-301}:\\
\;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2 \cdot a}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\

\mathbf{elif}\;b \le 9.748353205521284417498320088314179968082 \cdot 10^{77}:\\
\;\;\;\;\frac{\frac{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}\\

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

\end{array}
double f(double a, double b, double c) {
        double r150314 = b;
        double r150315 = -r150314;
        double r150316 = r150314 * r150314;
        double r150317 = 4.0;
        double r150318 = a;
        double r150319 = r150317 * r150318;
        double r150320 = c;
        double r150321 = r150319 * r150320;
        double r150322 = r150316 - r150321;
        double r150323 = sqrt(r150322);
        double r150324 = r150315 + r150323;
        double r150325 = 2.0;
        double r150326 = r150325 * r150318;
        double r150327 = r150324 / r150326;
        return r150327;
}
↓
double f(double a, double b, double c) {
        double r150328 = b;
        double r150329 = -2.5682011286372237e+55;
        bool r150330 = r150328 <= r150329;
        double r150331 = 1.0;
        double r150332 = c;
        double r150333 = r150332 / r150328;
        double r150334 = a;
        double r150335 = r150328 / r150334;
        double r150336 = r150333 - r150335;
        double r150337 = r150331 * r150336;
        double r150338 = -4.546128927832045e-301;
        bool r150339 = r150328 <= r150338;
        double r150340 = -r150328;
        double r150341 = r150328 * r150328;
        double r150342 = 4.0;
        double r150343 = r150342 * r150334;
        double r150344 = r150343 * r150332;
        double r150345 = r150341 - r150344;
        double r150346 = sqrt(r150345);
        double r150347 = r150340 + r150346;
        double r150348 = sqrt(r150347);
        double r150349 = 2.0;
        double r150350 = r150349 * r150334;
        double r150351 = r150350 / r150348;
        double r150352 = r150348 / r150351;
        double r150353 = 9.748353205521284e+77;
        bool r150354 = r150328 <= r150353;
        double r150355 = 0.0;
        double r150356 = r150334 * r150332;
        double r150357 = r150342 * r150356;
        double r150358 = r150355 + r150357;
        double r150359 = r150340 - r150346;
        double r150360 = r150358 / r150359;
        double r150361 = r150360 / r150350;
        double r150362 = -1.0;
        double r150363 = r150362 * r150333;
        double r150364 = r150354 ? r150361 : r150363;
        double r150365 = r150339 ? r150352 : r150364;
        double r150366 = r150330 ? r150337 : r150365;
        return r150366;
}

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
Target21.0
Herbie9.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.5682011286372237e+55

    1. Initial program 38.3

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

    2. Taylor expanded around -inf 5.4

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

    3. Simplified5.4

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

    if -2.5682011286372237e+55 < b < -4.546128927832045e-301

    1. Initial program 9.5

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

    3. Applied add-sqr-sqrt9.8

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

    4. Applied associate-/l*9.9

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

    if -4.546128927832045e-301 < b < 9.748353205521284e+77

    1. Initial program 31.3

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

    3. Applied flip-+31.4

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

    4. Simplified16.8

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

    if 9.748353205521284e+77 < b

    1. Initial program 58.3

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

    2. Taylor expanded around inf 2.8

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.568201128637223695690583924646661116118 \cdot 10^{55}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -4.546128927832045171835550197132658970541 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2 \cdot a}{\sqrt{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \mathbf{elif}\;b \le 9.748353205521284417498320088314179968082 \cdot 10^{77}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(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)))