Average Error: 34.4 → 10.0
Time: 15.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.297624534318876743725099723501638614139 \cdot 10^{152}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -2.297624534318876743725099723501638614139 \cdot 10^{152}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r67496 = b;
        double r67497 = -r67496;
        double r67498 = r67496 * r67496;
        double r67499 = 4.0;
        double r67500 = a;
        double r67501 = c;
        double r67502 = r67500 * r67501;
        double r67503 = r67499 * r67502;
        double r67504 = r67498 - r67503;
        double r67505 = sqrt(r67504);
        double r67506 = r67497 + r67505;
        double r67507 = 2.0;
        double r67508 = r67507 * r67500;
        double r67509 = r67506 / r67508;
        return r67509;
}
↓
double f(double a, double b, double c) {
        double r67510 = b;
        double r67511 = -2.2976245343188767e+152;
        bool r67512 = r67510 <= r67511;
        double r67513 = 1.0;
        double r67514 = c;
        double r67515 = r67514 / r67510;
        double r67516 = a;
        double r67517 = r67510 / r67516;
        double r67518 = r67515 - r67517;
        double r67519 = r67513 * r67518;
        double r67520 = 8.70366778308292e-52;
        bool r67521 = r67510 <= r67520;
        double r67522 = r67510 * r67510;
        double r67523 = 4.0;
        double r67524 = r67516 * r67514;
        double r67525 = r67523 * r67524;
        double r67526 = r67522 - r67525;
        double r67527 = sqrt(r67526);
        double r67528 = 2.0;
        double r67529 = r67528 * r67516;
        double r67530 = r67527 / r67529;
        double r67531 = r67510 / r67529;
        double r67532 = r67530 - r67531;
        double r67533 = -1.0;
        double r67534 = r67533 * r67515;
        double r67535 = r67521 ? r67532 : r67534;
        double r67536 = r67512 ? r67519 : r67535;
        return r67536;
}

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

Original34.4
Target21.4
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.2976245343188767e+152

    1. Initial program 63.3

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

    2. Simplified63.3

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

    3. Taylor expanded around -inf 2.2

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

    4. Simplified2.2

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

    if -2.2976245343188767e+152 < b < 8.70366778308292e-52

    1. Initial program 13.2

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

    2. Simplified13.2

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

    4. Applied div-sub13.2

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

    if 8.70366778308292e-52 < b

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded around inf 7.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.297624534318876743725099723501638614139 \cdot 10^{152}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.703667783082919749023199154845924676168 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 
(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)))