Average Error: 34.0 → 9.6
Time: 19.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3438870219673743856300090597329338368:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.940959811381675939988945669030883027544 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le -6.373481960958038092461518588786337642855 \cdot 10^{-132}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.972594060757836305384453450333248094101 \cdot 10^{74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -3438870219673743856300090597329338368:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -9.940959811381675939988945669030883027544 \cdot 10^{-96}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le -6.373481960958038092461518588786337642855 \cdot 10^{-132}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.972594060757836305384453450333248094101 \cdot 10^{74}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r105586 = b;
        double r105587 = -r105586;
        double r105588 = r105586 * r105586;
        double r105589 = 4.0;
        double r105590 = a;
        double r105591 = c;
        double r105592 = r105590 * r105591;
        double r105593 = r105589 * r105592;
        double r105594 = r105588 - r105593;
        double r105595 = sqrt(r105594);
        double r105596 = r105587 - r105595;
        double r105597 = 2.0;
        double r105598 = r105597 * r105590;
        double r105599 = r105596 / r105598;
        return r105599;
}
↓
double f(double a, double b, double c) {
        double r105600 = b;
        double r105601 = -3.438870219673744e+36;
        bool r105602 = r105600 <= r105601;
        double r105603 = -1.0;
        double r105604 = c;
        double r105605 = r105604 / r105600;
        double r105606 = r105603 * r105605;
        double r105607 = -9.940959811381676e-96;
        bool r105608 = r105600 <= r105607;
        double r105609 = 4.0;
        double r105610 = a;
        double r105611 = r105610 * r105604;
        double r105612 = r105609 * r105611;
        double r105613 = r105600 * r105600;
        double r105614 = r105613 - r105612;
        double r105615 = cbrt(r105614);
        double r105616 = r105615 * r105615;
        double r105617 = sqrt(r105616);
        double r105618 = sqrt(r105615);
        double r105619 = -r105600;
        double r105620 = fma(r105617, r105618, r105619);
        double r105621 = r105612 / r105620;
        double r105622 = 2.0;
        double r105623 = r105622 * r105610;
        double r105624 = r105621 / r105623;
        double r105625 = -6.373481960958038e-132;
        bool r105626 = r105600 <= r105625;
        double r105627 = 2.9725940607578363e+74;
        bool r105628 = r105600 <= r105627;
        double r105629 = r105609 * r105610;
        double r105630 = r105629 * r105604;
        double r105631 = r105613 - r105630;
        double r105632 = sqrt(r105631);
        double r105633 = r105619 - r105632;
        double r105634 = r105633 / r105623;
        double r105635 = 1.0;
        double r105636 = r105600 / r105610;
        double r105637 = r105605 - r105636;
        double r105638 = r105635 * r105637;
        double r105639 = r105628 ? r105634 : r105638;
        double r105640 = r105626 ? r105606 : r105639;
        double r105641 = r105608 ? r105624 : r105640;
        double r105642 = r105602 ? r105606 : r105641;
        return r105642;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.9
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.438870219673744e+36 or -9.940959811381676e-96 < b < -6.373481960958038e-132

    1. Initial program 54.0

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

    2. Taylor expanded around -inf 6.8

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

    if -3.438870219673744e+36 < b < -9.940959811381676e-96

    1. Initial program 41.3

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

    3. Applied flip--41.4

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

    4. Simplified17.1

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

    5. Simplified17.1

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

    7. Applied add-cube-cbrt17.4

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{\color{blue}{\left(\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} - b}}{2 \cdot a}\]

    8. Applied sqrt-prod17.3

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} - b}}{2 \cdot a}\]

    9. Applied fma-neg17.3

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}}{2 \cdot a}\]

    if -6.373481960958038e-132 < b < 2.9725940607578363e+74

    1. Initial program 11.6

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

    3. Applied associate-*r*11.6

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

    if 2.9725940607578363e+74 < b

    1. Initial program 41.8

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

    2. Taylor expanded around inf 5.3

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

    3. Simplified5.3

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3438870219673743856300090597329338368:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -9.940959811381675939988945669030883027544 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, \sqrt{\sqrt[3]{b \cdot b - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le -6.373481960958038092461518588786337642855 \cdot 10^{-132}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.972594060757836305384453450333248094101 \cdot 10^{74}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))