Average Error: 33.2 → 9.9
Time: 21.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1157171 = b_2;
        double r1157172 = -r1157171;
        double r1157173 = r1157171 * r1157171;
        double r1157174 = a;
        double r1157175 = c;
        double r1157176 = r1157174 * r1157175;
        double r1157177 = r1157173 - r1157176;
        double r1157178 = sqrt(r1157177);
        double r1157179 = r1157172 - r1157178;
        double r1157180 = r1157179 / r1157174;
        return r1157180;
}
↓
double f(double a, double b_2, double c) {
        double r1157181 = b_2;
        double r1157182 = -1.8774910265390396e-73;
        bool r1157183 = r1157181 <= r1157182;
        double r1157184 = -0.5;
        double r1157185 = c;
        double r1157186 = r1157185 / r1157181;
        double r1157187 = r1157184 * r1157186;
        double r1157188 = 2.5703497435733685e+102;
        bool r1157189 = r1157181 <= r1157188;
        double r1157190 = -r1157181;
        double r1157191 = r1157181 * r1157181;
        double r1157192 = a;
        double r1157193 = r1157192 * r1157185;
        double r1157194 = r1157191 - r1157193;
        double r1157195 = sqrt(r1157194);
        double r1157196 = r1157190 - r1157195;
        double r1157197 = 1.0;
        double r1157198 = r1157197 / r1157192;
        double r1157199 = r1157196 * r1157198;
        double r1157200 = 0.5;
        double r1157201 = -2.0;
        double r1157202 = r1157181 / r1157192;
        double r1157203 = r1157201 * r1157202;
        double r1157204 = fma(r1157186, r1157200, r1157203);
        double r1157205 = r1157189 ? r1157199 : r1157204;
        double r1157206 = r1157183 ? r1157187 : r1157205;
        return r1157206;
}

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 < -1.8774910265390396e-73

    1. Initial program 52.5

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

    3. Applied div-inv52.5

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

    4. Taylor expanded around -inf 8.6

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

    if -1.8774910265390396e-73 < b_2 < 2.5703497435733685e+102

    1. Initial program 13.1

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

    3. Applied div-inv13.2

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

    if 2.5703497435733685e+102 < b_2

    1. Initial program 43.9

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

    3. Applied div-inv44.0

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

    4. Taylor expanded around inf 2.9

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

    5. Simplified2.9

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))