Average Error: 33.8 → 10.2
Time: 19.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{b_2}{a} \cdot -2\\

\end{array}
double f(double a, double b_2, double c) {
        double r3175141 = b_2;
        double r3175142 = -r3175141;
        double r3175143 = r3175141 * r3175141;
        double r3175144 = a;
        double r3175145 = c;
        double r3175146 = r3175144 * r3175145;
        double r3175147 = r3175143 - r3175146;
        double r3175148 = sqrt(r3175147);
        double r3175149 = r3175142 - r3175148;
        double r3175150 = r3175149 / r3175144;
        return r3175150;
}
↓
double f(double a, double b_2, double c) {
        double r3175151 = b_2;
        double r3175152 = -2.5694949190681246e-64;
        bool r3175153 = r3175151 <= r3175152;
        double r3175154 = -0.5;
        double r3175155 = c;
        double r3175156 = r3175155 / r3175151;
        double r3175157 = r3175154 * r3175156;
        double r3175158 = 2.865381670376961e+117;
        bool r3175159 = r3175151 <= r3175158;
        double r3175160 = -r3175151;
        double r3175161 = r3175151 * r3175151;
        double r3175162 = a;
        double r3175163 = r3175162 * r3175155;
        double r3175164 = r3175161 - r3175163;
        double r3175165 = sqrt(r3175164);
        double r3175166 = r3175160 - r3175165;
        double r3175167 = r3175166 / r3175162;
        double r3175168 = r3175151 / r3175162;
        double r3175169 = -2.0;
        double r3175170 = r3175168 * r3175169;
        double r3175171 = r3175159 ? r3175167 : r3175170;
        double r3175172 = r3175153 ? r3175157 : r3175171;
        return r3175172;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.5694949190681246e-64

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 9.1

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

    if -2.5694949190681246e-64 < b_2 < 2.865381670376961e+117

    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

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

    5. Applied un-div-inv13.1

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

    if 2.865381670376961e+117 < b_2

    1. Initial program 52.0

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

    3. Applied clear-num52.1

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

    4. Taylor expanded around 0 3.1

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.569494919068124572690421335939486791404 \cdot 10^{-64}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8653816703769607550753035783606354728 \cdot 10^{117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))