Average Error: 34.3 → 10.4
Time: 38.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877097952776006549579654784856 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c \cdot \frac{1}{2}}{b_2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.264659490877097952776006549579654784856 \cdot 10^{-67}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\
\;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r2779149 = b_2;
        double r2779150 = -r2779149;
        double r2779151 = r2779149 * r2779149;
        double r2779152 = a;
        double r2779153 = c;
        double r2779154 = r2779152 * r2779153;
        double r2779155 = r2779151 - r2779154;
        double r2779156 = sqrt(r2779155);
        double r2779157 = r2779150 - r2779156;
        double r2779158 = r2779157 / r2779152;
        return r2779158;
}
↓
double f(double a, double b_2, double c) {
        double r2779159 = b_2;
        double r2779160 = -1.264659490877098e-67;
        bool r2779161 = r2779159 <= r2779160;
        double r2779162 = -0.5;
        double r2779163 = c;
        double r2779164 = r2779163 / r2779159;
        double r2779165 = r2779162 * r2779164;
        double r2779166 = 0.17389787404847717;
        bool r2779167 = r2779159 <= r2779166;
        double r2779168 = r2779159 * r2779159;
        double r2779169 = a;
        double r2779170 = r2779163 * r2779169;
        double r2779171 = r2779168 - r2779170;
        double r2779172 = sqrt(r2779171);
        double r2779173 = r2779172 + r2779159;
        double r2779174 = -r2779173;
        double r2779175 = r2779174 / r2779169;
        double r2779176 = r2779159 / r2779169;
        double r2779177 = -2.0;
        double r2779178 = 0.5;
        double r2779179 = r2779163 * r2779178;
        double r2779180 = r2779179 / r2779159;
        double r2779181 = fma(r2779176, r2779177, r2779180);
        double r2779182 = r2779167 ? r2779175 : r2779181;
        double r2779183 = r2779161 ? r2779165 : r2779182;
        return r2779183;
}

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 53.5

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

    2. Taylor expanded around -inf 8.1

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

    if -1.264659490877098e-67 < b_2 < 0.17389787404847717

    1. Initial program 15.2

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

    3. Applied *-un-lft-identity15.2

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

    4. Applied associate-/r*15.2

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

    5. Simplified15.2

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

    if 0.17389787404847717 < b_2

    1. Initial program 31.2

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

    3. Applied *-un-lft-identity31.2

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

    4. Applied associate-/r*31.2

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

    5. Simplified31.2

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

    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

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.264659490877097952776006549579654784856 \cdot 10^{-67}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 0.173897874048477174557802982235443778336:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c \cdot \frac{1}{2}}{b_2}\right)\\ \end{array}\]

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))