Average Error: 34.4 → 10.6
Time: 17.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r5673462 = b;
        double r5673463 = -r5673462;
        double r5673464 = r5673462 * r5673462;
        double r5673465 = 4.0;
        double r5673466 = a;
        double r5673467 = c;
        double r5673468 = r5673466 * r5673467;
        double r5673469 = r5673465 * r5673468;
        double r5673470 = r5673464 - r5673469;
        double r5673471 = sqrt(r5673470);
        double r5673472 = r5673463 + r5673471;
        double r5673473 = 2.0;
        double r5673474 = r5673473 * r5673466;
        double r5673475 = r5673472 / r5673474;
        return r5673475;
}
↓
double f(double a, double b, double c) {
        double r5673476 = b;
        double r5673477 = -2.221067196710922e+149;
        bool r5673478 = r5673476 <= r5673477;
        double r5673479 = 2.0;
        double r5673480 = c;
        double r5673481 = r5673480 / r5673476;
        double r5673482 = r5673479 * r5673481;
        double r5673483 = a;
        double r5673484 = r5673476 / r5673483;
        double r5673485 = 2.0;
        double r5673486 = r5673484 * r5673485;
        double r5673487 = r5673482 - r5673486;
        double r5673488 = r5673487 / r5673479;
        double r5673489 = 2.8983489306952693e-35;
        bool r5673490 = r5673476 <= r5673489;
        double r5673491 = r5673476 * r5673476;
        double r5673492 = 4.0;
        double r5673493 = r5673483 * r5673492;
        double r5673494 = r5673480 * r5673493;
        double r5673495 = r5673491 - r5673494;
        double r5673496 = sqrt(r5673495);
        double r5673497 = r5673496 - r5673476;
        double r5673498 = r5673497 / r5673483;
        double r5673499 = r5673498 / r5673479;
        double r5673500 = -2.0;
        double r5673501 = r5673500 * r5673481;
        double r5673502 = r5673501 / r5673479;
        double r5673503 = r5673490 ? r5673499 : r5673502;
        double r5673504 = r5673478 ? r5673488 : r5673503;
        return r5673504;
}

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
Target21.5
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    3. Taylor expanded around -inf 2.8

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.6

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

    2. Simplified14.6

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{2 \cdot \frac{c}{b} - \frac{b}{a} \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))