Average Error: 32.8 → 10.0
Time: 19.3s
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 r1226508 = b;
        double r1226509 = -r1226508;
        double r1226510 = r1226508 * r1226508;
        double r1226511 = 4.0;
        double r1226512 = a;
        double r1226513 = r1226511 * r1226512;
        double r1226514 = c;
        double r1226515 = r1226513 * r1226514;
        double r1226516 = r1226510 - r1226515;
        double r1226517 = sqrt(r1226516);
        double r1226518 = r1226509 + r1226517;
        double r1226519 = 2.0;
        double r1226520 = r1226519 * r1226512;
        double r1226521 = r1226518 / r1226520;
        return r1226521;
}
↓
double f(double a, double b, double c) {
        double r1226522 = b;
        double r1226523 = -3.063397748446981e+71;
        bool r1226524 = r1226522 <= r1226523;
        double r1226525 = c;
        double r1226526 = r1226525 / r1226522;
        double r1226527 = a;
        double r1226528 = r1226522 / r1226527;
        double r1226529 = r1226526 - r1226528;
        double r1226530 = 2.0;
        double r1226531 = r1226529 * r1226530;
        double r1226532 = r1226531 / r1226530;
        double r1226533 = 3.1295384133612364e-73;
        bool r1226534 = r1226522 <= r1226533;
        double r1226535 = 1.0;
        double r1226536 = r1226535 / r1226527;
        double r1226537 = -4.0;
        double r1226538 = r1226527 * r1226537;
        double r1226539 = r1226538 * r1226525;
        double r1226540 = fma(r1226522, r1226522, r1226539);
        double r1226541 = sqrt(r1226540);
        double r1226542 = r1226541 - r1226522;
        double r1226543 = r1226536 * r1226542;
        double r1226544 = r1226543 / r1226530;
        double r1226545 = -2.0;
        double r1226546 = r1226545 * r1226526;
        double r1226547 = r1226546 / r1226530;
        double r1226548 = r1226534 ? r1226544 : r1226547;
        double r1226549 = r1226524 ? r1226532 : r1226548;
        return r1226549;
}

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)))