Average Error: 33.6 → 11.2
Time: 19.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.786204067849289 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.2890050783826923 \cdot 10^{-183}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.786204067849289 \cdot 10^{+100}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{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 r336097 = b_2;
        double r336098 = -r336097;
        double r336099 = r336097 * r336097;
        double r336100 = a;
        double r336101 = c;
        double r336102 = r336100 * r336101;
        double r336103 = r336099 - r336102;
        double r336104 = sqrt(r336103);
        double r336105 = r336098 - r336104;
        double r336106 = r336105 / r336100;
        return r336106;
}
↓
double f(double a, double b_2, double c) {
        double r336107 = b_2;
        double r336108 = -1.2890050783826923e-183;
        bool r336109 = r336107 <= r336108;
        double r336110 = -0.5;
        double r336111 = c;
        double r336112 = r336111 / r336107;
        double r336113 = r336110 * r336112;
        double r336114 = 1.786204067849289e+100;
        bool r336115 = r336107 <= r336114;
        double r336116 = -r336107;
        double r336117 = r336107 * r336107;
        double r336118 = a;
        double r336119 = r336118 * r336111;
        double r336120 = r336117 - r336119;
        double r336121 = sqrt(r336120);
        double r336122 = r336116 - r336121;
        double r336123 = r336122 / r336118;
        double r336124 = r336107 / r336118;
        double r336125 = -2.0;
        double r336126 = 0.5;
        double r336127 = r336107 / r336111;
        double r336128 = r336126 / r336127;
        double r336129 = fma(r336124, r336125, r336128);
        double r336130 = r336115 ? r336123 : r336129;
        double r336131 = r336109 ? r336113 : r336130;
        return r336131;
}

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 < -1.2890050783826923e-183

    1. Initial program 48.3

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

    2. Taylor expanded around inf 48.3

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified48.3

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

    4. Taylor expanded around -inf 14.3

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

    if -1.2890050783826923e-183 < b_2 < 1.786204067849289e+100

    1. Initial program 10.5

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

    2. Taylor expanded around inf 10.5

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{{b_2}^{2} - a \cdot c}}}{a}\]

    3. Simplified10.5

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

    if 1.786204067849289e+100 < b_2

    1. Initial program 44.2

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

    2. Taylor expanded around inf 3.4

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

    3. Simplified3.4

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

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.2890050783826923 \cdot 10^{-183}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.786204067849289 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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))