Average Error: 1.6 → 0.6
Time: 22.3s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.059906005859375:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}
↓
\begin{array}{l}
\mathbf{if}\;b \le -0.059906005859375:\\
\;\;\;\;\frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r864613 = b;
        double r864614 = -r864613;
        double r864615 = r864613 * r864613;
        double r864616 = 4.0;
        double r864617 = /* ERROR: no posit support in C */;
        double r864618 = a;
        double r864619 = c;
        double r864620 = r864618 * r864619;
        double r864621 = r864617 * r864620;
        double r864622 = r864615 - r864621;
        double r864623 = sqrt(r864622);
        double r864624 = r864614 - r864623;
        double r864625 = 2.0;
        double r864626 = /* ERROR: no posit support in C */;
        double r864627 = r864626 * r864618;
        double r864628 = r864624 / r864627;
        return r864628;
}
↓
double f(double a, double b, double c) {
        double r864629 = b;
        double r864630 = -0.059906005859375;
        bool r864631 = r864629 <= r864630;
        double r864632 = -r864629;
        double r864633 = r864632 + r864629;
        double r864634 = r864632 + r864632;
        double r864635 = r864633 * r864634;
        double r864636 = 4.0;
        double r864637 = a;
        double r864638 = c;
        double r864639 = r864637 * r864638;
        double r864640 = r864636 * r864639;
        double r864641 = r864635 + r864640;
        double r864642 = 2.0;
        double r864643 = r864642 * r864637;
        double r864644 = r864641 / r864643;
        double r864645 = r864629 * r864629;
        double r864646 = r864645 - r864640;
        double r864647 = sqrt(r864646);
        double r864648 = r864632 + r864647;
        double r864649 = r864644 / r864648;
        double r864650 = r864636 * r864637;
        double r864651 = r864650 * r864638;
        double r864652 = r864645 - r864651;
        double r864653 = sqrt(r864652);
        double r864654 = r864632 - r864653;
        double r864655 = r864654 / r864643;
        double r864656 = r864631 ? r864649 : r864655;
        return r864656;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b < -0.059906005859375

    1. Initial program 2.9

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

    3. Applied p16-flip--2.6

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

    4. Applied associate-/l/2.7

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

    5. Simplified0.8

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

    7. Applied associate-/r*0.7

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

    if -0.059906005859375 < b

    1. Initial program 0.5

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

    3. Applied associate-*r*0.5

      \[\leadsto \frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \color{blue}{\left(\left(\left(4\right) \cdot a\right) \cdot c\right)}\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.059906005859375:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) + b\right) \cdot \left(\left(-b\right) + \left(-b\right)\right) + 4 \cdot \left(a \cdot c\right)}{2 \cdot a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  (/.p16 (-.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))