Average Error: 1.7 → 0.5
Time: 16.5s
Precision: 64
\[\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -0.3563232421875:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\frac{b_2 \cdot b_2 + a \cdot c}{\frac{b_2 \cdot b_2 + a \cdot c}{b_2 \cdot b_2 - a \cdot c}}}}{a}\\ \end{array}\]
\frac{\left(\left(-b_2\right) - \left(\sqrt{\left(\left(b_2 \cdot b_2\right) - \left(a \cdot c\right)\right)}\right)\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -0.3563232421875:\\
\;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r730084 = b_2;
        double r730085 = -r730084;
        double r730086 = r730084 * r730084;
        double r730087 = a;
        double r730088 = c;
        double r730089 = r730087 * r730088;
        double r730090 = r730086 - r730089;
        double r730091 = sqrt(r730090);
        double r730092 = r730085 - r730091;
        double r730093 = r730092 / r730087;
        return r730093;
}
↓
double f(double a, double b_2, double c) {
        double r730094 = b_2;
        double r730095 = -0.3563232421875;
        bool r730096 = r730094 <= r730095;
        double r730097 = a;
        double r730098 = c;
        double r730099 = r730097 * r730098;
        double r730100 = -r730094;
        double r730101 = r730100 + r730094;
        double r730102 = r730100 + r730100;
        double r730103 = r730101 * r730102;
        double r730104 = r730099 + r730103;
        double r730105 = r730104 / r730097;
        double r730106 = r730094 * r730094;
        double r730107 = r730106 - r730099;
        double r730108 = sqrt(r730107);
        double r730109 = r730100 + r730108;
        double r730110 = r730105 / r730109;
        double r730111 = r730106 + r730099;
        double r730112 = r730111 / r730107;
        double r730113 = r730111 / r730112;
        double r730114 = sqrt(r730113);
        double r730115 = r730100 - r730114;
        double r730116 = r730115 / r730097;
        double r730117 = r730096 ? r730110 : r730116;
        return r730117;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 2 regimes

  2. if b_2 < -0.3563232421875

    1. Initial program 4.1

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

    3. Applied p16-flip--3.6

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

    4. Applied associate-/l/3.6

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

    5. Simplified0.7

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

    7. Applied associate-/r*0.5

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

    if -0.3563232421875 < b_2

    1. Initial program 0.4

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

    3. Applied p16-flip--1.3

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

    5. Applied difference-of-squares1.2

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

    6. Applied associate-/l*0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -0.3563232421875:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) + \left(-b_2\right)\right)}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\frac{b_2 \cdot b_2 + a \cdot c}{\frac{b_2 \cdot b_2 + a \cdot c}{b_2 \cdot b_2 - a \cdot c}}}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/.p16 (-.p16 (neg.p16 b_2) (sqrt.p16 (-.p16 (*.p16 b_2 b_2) (*.p16 a c)))) a))