Average Error: 34.1 → 9.2
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)}^{1}\\ \mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.3044033969831823 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.9238883452280037 \cdot 10^{-130}:\\
\;\;\;\;{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)}^{1}\\

\mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\
\;\;\;\;\frac{\frac{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}\\

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

\end{array}
double f(double a, double b, double c) {
        double r79394 = b;
        double r79395 = -r79394;
        double r79396 = r79394 * r79394;
        double r79397 = 4.0;
        double r79398 = a;
        double r79399 = c;
        double r79400 = r79398 * r79399;
        double r79401 = r79397 * r79400;
        double r79402 = r79396 - r79401;
        double r79403 = sqrt(r79402);
        double r79404 = r79395 + r79403;
        double r79405 = 2.0;
        double r79406 = r79405 * r79398;
        double r79407 = r79404 / r79406;
        return r79407;
}
↓
double f(double a, double b, double c) {
        double r79408 = b;
        double r79409 = -2.3044033969831823e+153;
        bool r79410 = r79408 <= r79409;
        double r79411 = 1.0;
        double r79412 = c;
        double r79413 = r79412 / r79408;
        double r79414 = a;
        double r79415 = r79408 / r79414;
        double r79416 = r79413 - r79415;
        double r79417 = r79411 * r79416;
        double r79418 = 1.9238883452280037e-130;
        bool r79419 = r79408 <= r79418;
        double r79420 = -r79408;
        double r79421 = r79408 * r79408;
        double r79422 = 4.0;
        double r79423 = r79414 * r79412;
        double r79424 = r79422 * r79423;
        double r79425 = r79421 - r79424;
        double r79426 = sqrt(r79425);
        double r79427 = r79420 + r79426;
        double r79428 = 2.0;
        double r79429 = r79428 * r79414;
        double r79430 = r79427 / r79429;
        double r79431 = 1.0;
        double r79432 = pow(r79430, r79431);
        double r79433 = 4.019930844191633e+109;
        bool r79434 = r79408 <= r79433;
        double r79435 = 0.0;
        double r79436 = r79435 + r79424;
        double r79437 = r79420 - r79426;
        double r79438 = r79436 / r79437;
        double r79439 = r79438 / r79429;
        double r79440 = -1.0;
        double r79441 = r79440 * r79413;
        double r79442 = r79434 ? r79439 : r79441;
        double r79443 = r79419 ? r79432 : r79442;
        double r79444 = r79410 ? r79417 : r79443;
        return r79444;
}

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

Original34.1
Target20.6
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.3044033969831823e+153

    1. Initial program 63.5

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -2.3044033969831823e+153 < b < 1.9238883452280037e-130

    1. Initial program 11.3

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

    3. Applied pow111.3

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

    if 1.9238883452280037e-130 < b < 4.019930844191633e+109

    1. Initial program 40.3

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

    3. Applied flip-+40.3

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

    4. Simplified15.5

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

    if 4.019930844191633e+109 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.4

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3044033969831823 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.9238883452280037 \cdot 10^{-130}:\\ \;\;\;\;{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\right)}^{1}\\ \mathbf{elif}\;b \le 4.01993084419163312 \cdot 10^{109}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))