Average Error: 34.0 → 10.5
Time: 35.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.229142930221511 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.229142930221511 \cdot 10^{-57}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.6656023684116586 \cdot 10^{+55}:\\
\;\;\;\;\left(-\frac{b_2}{a}\right) - \sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\

\end{array}
double f(double a, double b_2, double c) {
        double r796137 = b_2;
        double r796138 = -r796137;
        double r796139 = r796137 * r796137;
        double r796140 = a;
        double r796141 = c;
        double r796142 = r796140 * r796141;
        double r796143 = r796139 - r796142;
        double r796144 = sqrt(r796143);
        double r796145 = r796138 - r796144;
        double r796146 = r796145 / r796140;
        return r796146;
}
↓
double f(double a, double b_2, double c) {
        double r796147 = b_2;
        double r796148 = -1.229142930221511e-57;
        bool r796149 = r796147 <= r796148;
        double r796150 = -0.5;
        double r796151 = c;
        double r796152 = r796151 / r796147;
        double r796153 = r796150 * r796152;
        double r796154 = 2.6656023684116586e+55;
        bool r796155 = r796147 <= r796154;
        double r796156 = a;
        double r796157 = r796147 / r796156;
        double r796158 = -r796157;
        double r796159 = r796147 * r796147;
        double r796160 = r796151 * r796156;
        double r796161 = r796159 - r796160;
        double r796162 = sqrt(r796161);
        double r796163 = 1.0;
        double r796164 = r796163 / r796156;
        double r796165 = r796162 * r796164;
        double r796166 = r796158 - r796165;
        double r796167 = -2.0;
        double r796168 = 0.5;
        double r796169 = r796147 / r796151;
        double r796170 = r796168 / r796169;
        double r796171 = fma(r796157, r796167, r796170);
        double r796172 = r796155 ? r796166 : r796171;
        double r796173 = r796149 ? r796153 : r796172;
        return r796173;
}

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 < -1.229142930221511e-57

    1. Initial program 53.7

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

    3. Applied div-sub54.1

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

    4. Taylor expanded around -inf 8.2

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

    if -1.229142930221511e-57 < b_2 < 2.6656023684116586e+55

    1. Initial program 14.5

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

    3. Applied div-sub14.5

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

    5. Applied div-inv14.6

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

    if 2.6656023684116586e+55 < b_2

    1. Initial program 37.1

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

    2. Taylor expanded around inf 6.2

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

    3. Simplified6.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.229142930221511 \cdot 10^{-57}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.6656023684116586 \cdot 10^{+55}:\\ \;\;\;\;\left(-\frac{b_2}{a}\right) - \sqrt{b_2 \cdot b_2 - c \cdot a} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))