Average Error: 33.7 → 9.1
Time: 23.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.6806111715441095 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.5349830112643849 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 5.9911994584698608 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.6806111715441095 \cdot 10^{-29}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -2.5349830112643849 \cdot 10^{-83}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 5.9911994584698608 \cdot 10^{103}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r65034 = b_2;
        double r65035 = -r65034;
        double r65036 = r65034 * r65034;
        double r65037 = a;
        double r65038 = c;
        double r65039 = r65037 * r65038;
        double r65040 = r65036 - r65039;
        double r65041 = sqrt(r65040);
        double r65042 = r65035 - r65041;
        double r65043 = r65042 / r65037;
        return r65043;
}
↓
double f(double a, double b_2, double c) {
        double r65044 = b_2;
        double r65045 = -1.6806111715441095e-29;
        bool r65046 = r65044 <= r65045;
        double r65047 = -0.5;
        double r65048 = c;
        double r65049 = r65048 / r65044;
        double r65050 = r65047 * r65049;
        double r65051 = -2.534983011264385e-83;
        bool r65052 = r65044 <= r65051;
        double r65053 = a;
        double r65054 = r65053 * r65048;
        double r65055 = -r65048;
        double r65056 = r65044 * r65044;
        double r65057 = fma(r65055, r65053, r65056);
        double r65058 = sqrt(r65057);
        double r65059 = r65058 - r65044;
        double r65060 = r65054 / r65059;
        double r65061 = r65060 / r65053;
        double r65062 = 5.991199458469861e+103;
        bool r65063 = r65044 <= r65062;
        double r65064 = 1.0;
        double r65065 = -r65044;
        double r65066 = 2.0;
        double r65067 = pow(r65044, r65066);
        double r65068 = r65048 * r65053;
        double r65069 = r65067 - r65068;
        double r65070 = sqrt(r65069);
        double r65071 = r65065 - r65070;
        double r65072 = r65053 / r65071;
        double r65073 = r65064 / r65072;
        double r65074 = 0.5;
        double r65075 = -2.0;
        double r65076 = r65044 / r65053;
        double r65077 = r65075 * r65076;
        double r65078 = fma(r65074, r65049, r65077);
        double r65079 = r65063 ? r65073 : r65078;
        double r65080 = r65052 ? r65061 : r65079;
        double r65081 = r65046 ? r65050 : r65080;
        return r65081;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.6806111715441095e-29

    1. Initial program 54.8

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

    2. Taylor expanded around -inf 6.6

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

    if -1.6806111715441095e-29 < b_2 < -2.534983011264385e-83

    1. Initial program 32.9

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

    3. Applied flip--33.0

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

    4. Simplified16.9

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified16.9

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}}{a}\]

    if -2.534983011264385e-83 < b_2 < 5.991199458469861e+103

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    4. Simplified12.1

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

    if 5.991199458469861e+103 < b_2

    1. Initial program 48.7

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.6806111715441095 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -2.5349830112643849 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 5.9911994584698608 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{{b_2}^{2} - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))