Average Error: 32.8 → 10.0
Time: 17.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -3.063397748446981 \cdot 10^{+71}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1693419 = b;
        double r1693420 = -r1693419;
        double r1693421 = r1693419 * r1693419;
        double r1693422 = 4.0;
        double r1693423 = a;
        double r1693424 = r1693422 * r1693423;
        double r1693425 = c;
        double r1693426 = r1693424 * r1693425;
        double r1693427 = r1693421 - r1693426;
        double r1693428 = sqrt(r1693427);
        double r1693429 = r1693420 + r1693428;
        double r1693430 = 2.0;
        double r1693431 = r1693430 * r1693423;
        double r1693432 = r1693429 / r1693431;
        return r1693432;
}
↓
double f(double a, double b, double c) {
        double r1693433 = b;
        double r1693434 = -3.063397748446981e+71;
        bool r1693435 = r1693433 <= r1693434;
        double r1693436 = c;
        double r1693437 = r1693436 / r1693433;
        double r1693438 = a;
        double r1693439 = r1693433 / r1693438;
        double r1693440 = r1693437 - r1693439;
        double r1693441 = 2.0;
        double r1693442 = r1693440 * r1693441;
        double r1693443 = r1693442 / r1693441;
        double r1693444 = 3.1295384133612364e-73;
        bool r1693445 = r1693433 <= r1693444;
        double r1693446 = 1.0;
        double r1693447 = r1693446 / r1693438;
        double r1693448 = -4.0;
        double r1693449 = r1693438 * r1693448;
        double r1693450 = r1693449 * r1693436;
        double r1693451 = fma(r1693433, r1693433, r1693450);
        double r1693452 = sqrt(r1693451);
        double r1693453 = r1693452 - r1693433;
        double r1693454 = r1693447 * r1693453;
        double r1693455 = r1693454 / r1693441;
        double r1693456 = -2.0;
        double r1693457 = r1693456 * r1693437;
        double r1693458 = r1693457 / r1693441;
        double r1693459 = r1693445 ? r1693455 : r1693458;
        double r1693460 = r1693435 ? r1693443 : r1693459;
        return r1693460;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.8
Target20.1
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.063397748446981e+71

    1. Initial program 38.6

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

    2. Simplified38.6

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

    4. Applied div-inv38.7

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

    5. Taylor expanded around -inf 4.7

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

    6. Simplified4.7

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

    if -3.063397748446981e+71 < b < 3.1295384133612364e-73

    1. Initial program 13.0

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

    2. Simplified13.0

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

    4. Applied div-inv13.2

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

    if 3.1295384133612364e-73 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.063397748446981 \cdot 10^{+71}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.1295384133612364 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))