Average Error: 33.6 → 10.3
Time: 18.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.264659490877098 \cdot 10^{-67}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 0.17389787404847717:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r3130656 = b_2;
        double r3130657 = -r3130656;
        double r3130658 = r3130656 * r3130656;
        double r3130659 = a;
        double r3130660 = c;
        double r3130661 = r3130659 * r3130660;
        double r3130662 = r3130658 - r3130661;
        double r3130663 = sqrt(r3130662);
        double r3130664 = r3130657 - r3130663;
        double r3130665 = r3130664 / r3130659;
        return r3130665;
}
↓
double f(double a, double b_2, double c) {
        double r3130666 = b_2;
        double r3130667 = -1.264659490877098e-67;
        bool r3130668 = r3130666 <= r3130667;
        double r3130669 = -0.5;
        double r3130670 = c;
        double r3130671 = r3130670 / r3130666;
        double r3130672 = r3130669 * r3130671;
        double r3130673 = 0.17389787404847717;
        bool r3130674 = r3130666 <= r3130673;
        double r3130675 = -r3130666;
        double r3130676 = r3130666 * r3130666;
        double r3130677 = a;
        double r3130678 = r3130677 * r3130670;
        double r3130679 = r3130676 - r3130678;
        double r3130680 = sqrt(r3130679);
        double r3130681 = r3130675 - r3130680;
        double r3130682 = r3130681 / r3130677;
        double r3130683 = r3130666 / r3130677;
        double r3130684 = -2.0;
        double r3130685 = 2.0;
        double r3130686 = r3130671 / r3130685;
        double r3130687 = fma(r3130683, r3130684, r3130686);
        double r3130688 = r3130674 ? r3130682 : r3130687;
        double r3130689 = r3130668 ? r3130672 : r3130688;
        return r3130689;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -1.264659490877098e-67

    1. Initial program 53.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 8.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -1.264659490877098e-67 < b_2 < 0.17389787404847717

    1. Initial program 15.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv15.1

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied un-div-inv15.0

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    if 0.17389787404847717 < b_2

    1. Initial program 29.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv29.9

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied un-div-inv29.8

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}}\]

    6. Taylor expanded around inf 7.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    7. Simplified7.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877098 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.17389787404847717:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{c}{b_2}}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))