Average Error: 32.9 → 10.4
Time: 1.1m
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.805535571809126 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.4120493174601926 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(b_2 \cdot \frac{-2}{a}\right)\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.805535571809126 \cdot 10^{-39}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.4120493174601926 \cdot 10^{+147}:\\
\;\;\;\;\frac{1}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r1752086 = b_2;
        double r1752087 = -r1752086;
        double r1752088 = r1752086 * r1752086;
        double r1752089 = a;
        double r1752090 = c;
        double r1752091 = r1752089 * r1752090;
        double r1752092 = r1752088 - r1752091;
        double r1752093 = sqrt(r1752092);
        double r1752094 = r1752087 - r1752093;
        double r1752095 = r1752094 / r1752089;
        return r1752095;
}
↓
double f(double a, double b_2, double c) {
        double r1752096 = b_2;
        double r1752097 = -3.805535571809126e-39;
        bool r1752098 = r1752096 <= r1752097;
        double r1752099 = -0.5;
        double r1752100 = c;
        double r1752101 = r1752100 / r1752096;
        double r1752102 = r1752099 * r1752101;
        double r1752103 = 3.4120493174601926e+147;
        bool r1752104 = r1752096 <= r1752103;
        double r1752105 = 1.0;
        double r1752106 = a;
        double r1752107 = -r1752096;
        double r1752108 = r1752096 * r1752096;
        double r1752109 = r1752100 * r1752106;
        double r1752110 = r1752108 - r1752109;
        double r1752111 = sqrt(r1752110);
        double r1752112 = r1752107 - r1752111;
        double r1752113 = r1752105 / r1752112;
        double r1752114 = r1752106 * r1752113;
        double r1752115 = r1752105 / r1752114;
        double r1752116 = 0.5;
        double r1752117 = -2.0;
        double r1752118 = r1752117 / r1752106;
        double r1752119 = r1752096 * r1752118;
        double r1752120 = fma(r1752101, r1752116, r1752119);
        double r1752121 = r1752104 ? r1752115 : r1752120;
        double r1752122 = r1752098 ? r1752102 : r1752121;
        return r1752122;
}

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 < -3.805535571809126e-39

    1. Initial program 53.6

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

    2. Taylor expanded around -inf 7.7

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

    if -3.805535571809126e-39 < b_2 < 3.4120493174601926e+147

    1. Initial program 13.5

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

    3. Applied *-un-lft-identity13.5

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

    4. Applied associate-/l*13.6

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

    6. Applied div-inv13.7

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

    if 3.4120493174601926e+147 < b_2

    1. Initial program 58.6

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

    2. Taylor expanded around inf 3.3

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

    3. Simplified3.5

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.805535571809126 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.4120493174601926 \cdot 10^{+147}:\\ \;\;\;\;\frac{1}{a \cdot \frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{c}{b_2}\right), \frac{1}{2}, \left(b_2 \cdot \frac{-2}{a}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +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))