Average Error: 33.2 → 10.3
Time: 28.8s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.571206846913461 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.571206846913461 \cdot 10^{+106}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 3.821014310434392 \cdot 10^{-21}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r615109 = b_2;
        double r615110 = -r615109;
        double r615111 = r615109 * r615109;
        double r615112 = a;
        double r615113 = c;
        double r615114 = r615112 * r615113;
        double r615115 = r615111 - r615114;
        double r615116 = sqrt(r615115);
        double r615117 = r615110 + r615116;
        double r615118 = r615117 / r615112;
        return r615118;
}
↓
double f(double a, double b_2, double c) {
        double r615119 = b_2;
        double r615120 = -5.571206846913461e+106;
        bool r615121 = r615119 <= r615120;
        double r615122 = a;
        double r615123 = r615119 / r615122;
        double r615124 = -2.0;
        double r615125 = 0.5;
        double r615126 = c;
        double r615127 = r615126 / r615119;
        double r615128 = r615125 * r615127;
        double r615129 = fma(r615123, r615124, r615128);
        double r615130 = 3.821014310434392e-21;
        bool r615131 = r615119 <= r615130;
        double r615132 = r615119 * r615119;
        double r615133 = r615122 * r615126;
        double r615134 = r615132 - r615133;
        double r615135 = sqrt(r615134);
        double r615136 = r615135 - r615119;
        double r615137 = r615136 / r615122;
        double r615138 = -0.5;
        double r615139 = r615127 * r615138;
        double r615140 = r615131 ? r615137 : r615139;
        double r615141 = r615121 ? r615129 : r615140;
        return r615141;
}

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 < -5.571206846913461e+106

    1. Initial program 46.4

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

    2. Simplified46.4

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

    3. Taylor expanded around inf 46.4

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

    4. Simplified46.4

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

    5. Taylor expanded around -inf 3.6

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

    6. Simplified3.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)}\]

    if -5.571206846913461e+106 < b_2 < 3.821014310434392e-21

    1. Initial program 14.7

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

    2. Simplified14.7

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

    3. Taylor expanded around inf 14.7

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

    4. Simplified14.7

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

    if 3.821014310434392e-21 < b_2

    1. Initial program 54.7

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

    2. Simplified54.7

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

    3. Taylor expanded around inf 54.7

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

    4. Simplified54.7

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

    5. Taylor expanded around inf 6.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.571206846913461 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 3.821014310434392 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))