Average Error: 34.4 → 9.0
Time: 20.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.532647813487865 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.571246895950166 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 6.63703793523525319 \cdot 10^{98}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a} - \left(\frac{b_2}{a} - \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.532647813487865 \cdot 10^{52}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.571246895950166 \cdot 10^{-147}:\\
\;\;\;\;\frac{\frac{c \cdot a}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le 6.63703793523525319 \cdot 10^{98}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r79075 = b_2;
        double r79076 = -r79075;
        double r79077 = r79075 * r79075;
        double r79078 = a;
        double r79079 = c;
        double r79080 = r79078 * r79079;
        double r79081 = r79077 - r79080;
        double r79082 = sqrt(r79081);
        double r79083 = r79076 - r79082;
        double r79084 = r79083 / r79078;
        return r79084;
}
↓
double f(double a, double b_2, double c) {
        double r79085 = b_2;
        double r79086 = -1.532647813487865e+52;
        bool r79087 = r79085 <= r79086;
        double r79088 = -0.5;
        double r79089 = c;
        double r79090 = r79089 / r79085;
        double r79091 = r79088 * r79090;
        double r79092 = -1.571246895950166e-147;
        bool r79093 = r79085 <= r79092;
        double r79094 = a;
        double r79095 = r79089 * r79094;
        double r79096 = -r79095;
        double r79097 = fma(r79085, r79085, r79096);
        double r79098 = sqrt(r79097);
        double r79099 = r79098 - r79085;
        double r79100 = r79095 / r79099;
        double r79101 = r79100 / r79094;
        double r79102 = 6.637037935235253e+98;
        bool r79103 = r79085 <= r79102;
        double r79104 = -r79085;
        double r79105 = r79104 / r79094;
        double r79106 = r79085 * r79085;
        double r79107 = r79094 * r79089;
        double r79108 = r79106 - r79107;
        double r79109 = sqrt(r79108);
        double r79110 = r79109 / r79094;
        double r79111 = r79105 - r79110;
        double r79112 = r79085 / r79094;
        double r79113 = 0.5;
        double r79114 = r79113 * r79090;
        double r79115 = r79112 - r79114;
        double r79116 = r79105 - r79115;
        double r79117 = r79103 ? r79111 : r79116;
        double r79118 = r79093 ? r79101 : r79117;
        double r79119 = r79087 ? r79091 : r79118;
        return r79119;
}

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 < -1.532647813487865e+52

    1. Initial program 57.3

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

    2. Taylor expanded around -inf 3.6

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

    if -1.532647813487865e+52 < b_2 < -1.571246895950166e-147

    1. Initial program 37.3

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

    3. Applied flip--37.3

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

    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

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

    if -1.571246895950166e-147 < b_2 < 6.637037935235253e+98

    1. Initial program 11.7

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

    3. Applied div-sub11.7

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

    if 6.637037935235253e+98 < b_2

    1. Initial program 47.8

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

    3. Applied div-sub47.8

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

    4. Taylor expanded around inf 3.7

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.532647813487865 \cdot 10^{52}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.571246895950166 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le 6.63703793523525319 \cdot 10^{98}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b_2}{a} - \left(\frac{b_2}{a} - \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))