Average Error: 34.0 → 9.9
Time: 11.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.1973887031618163923063491438369965180755:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.724197465942408751027868496272110613308 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 813278.22458350402303040027618408203125:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -0.1973887031618163923063491438369965180755:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.724197465942408751027868496272110613308 \cdot 10^{-216}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le 813278.22458350402303040027618408203125:\\
\;\;\;\;\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 r41508 = b;
        double r41509 = -r41508;
        double r41510 = r41508 * r41508;
        double r41511 = 4.0;
        double r41512 = a;
        double r41513 = c;
        double r41514 = r41512 * r41513;
        double r41515 = r41511 * r41514;
        double r41516 = r41510 - r41515;
        double r41517 = sqrt(r41516);
        double r41518 = r41509 + r41517;
        double r41519 = 2.0;
        double r41520 = r41519 * r41512;
        double r41521 = r41518 / r41520;
        return r41521;
}
↓
double f(double a, double b, double c) {
        double r41522 = b;
        double r41523 = -0.1973887031618164;
        bool r41524 = r41522 <= r41523;
        double r41525 = 1.0;
        double r41526 = c;
        double r41527 = r41526 / r41522;
        double r41528 = a;
        double r41529 = r41522 / r41528;
        double r41530 = r41527 - r41529;
        double r41531 = r41525 * r41530;
        double r41532 = 1.7241974659424088e-216;
        bool r41533 = r41522 <= r41532;
        double r41534 = -r41522;
        double r41535 = r41522 * r41522;
        double r41536 = 4.0;
        double r41537 = r41528 * r41526;
        double r41538 = r41536 * r41537;
        double r41539 = r41535 - r41538;
        double r41540 = sqrt(r41539);
        double r41541 = r41534 + r41540;
        double r41542 = 2.0;
        double r41543 = r41542 * r41528;
        double r41544 = r41541 / r41543;
        double r41545 = 813278.224583504;
        bool r41546 = r41522 <= r41545;
        double r41547 = 0.0;
        double r41548 = r41547 + r41538;
        double r41549 = r41534 - r41540;
        double r41550 = r41548 / r41549;
        double r41551 = r41550 / r41543;
        double r41552 = -1.0;
        double r41553 = r41552 * r41527;
        double r41554 = r41546 ? r41551 : r41553;
        double r41555 = r41533 ? r41544 : r41554;
        double r41556 = r41524 ? r41531 : r41555;
        return r41556;
}

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

Original34.0
Target21.4
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -0.1973887031618164

    1. Initial program 32.2

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

    2. Taylor expanded around -inf 8.0

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

    3. Simplified8.0

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

    if -0.1973887031618164 < b < 1.7241974659424088e-216

    1. Initial program 11.8

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

    if 1.7241974659424088e-216 < b < 813278.224583504

    1. Initial program 28.8

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

    3. Applied flip-+28.9

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

    4. Simplified17.9

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

    if 813278.224583504 < b

    1. Initial program 55.9

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

    2. Taylor expanded around inf 5.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.1973887031618163923063491438369965180755:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.724197465942408751027868496272110613308 \cdot 10^{-216}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le 813278.22458350402303040027618408203125:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(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)))