Average Error: 33.6 → 9.8
Time: 19.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.634898599408338 \cdot 10^{+146}:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.1855168042470635 \cdot 10^{-53}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.634898599408338 \cdot 10^{+146}:\\
\;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r2477084 = b_2;
        double r2477085 = -r2477084;
        double r2477086 = r2477084 * r2477084;
        double r2477087 = a;
        double r2477088 = c;
        double r2477089 = r2477087 * r2477088;
        double r2477090 = r2477086 - r2477089;
        double r2477091 = sqrt(r2477090);
        double r2477092 = r2477085 - r2477091;
        double r2477093 = r2477092 / r2477087;
        return r2477093;
}
↓
double f(double a, double b_2, double c) {
        double r2477094 = b_2;
        double r2477095 = -8.1855168042470635e-53;
        bool r2477096 = r2477094 <= r2477095;
        double r2477097 = -0.5;
        double r2477098 = c;
        double r2477099 = r2477098 / r2477094;
        double r2477100 = r2477097 * r2477099;
        double r2477101 = 3.634898599408338e+146;
        bool r2477102 = r2477094 <= r2477101;
        double r2477103 = r2477094 * r2477094;
        double r2477104 = a;
        double r2477105 = r2477098 * r2477104;
        double r2477106 = r2477103 - r2477105;
        double r2477107 = sqrt(r2477106);
        double r2477108 = r2477107 + r2477094;
        double r2477109 = -r2477108;
        double r2477110 = r2477109 / r2477104;
        double r2477111 = 0.5;
        double r2477112 = r2477094 / r2477104;
        double r2477113 = -2.0;
        double r2477114 = r2477112 * r2477113;
        double r2477115 = fma(r2477099, r2477111, r2477114);
        double r2477116 = r2477102 ? r2477110 : r2477115;
        double r2477117 = r2477096 ? r2477100 : r2477116;
        return r2477117;
}

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 < -8.1855168042470635e-53

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 7.5

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

    if -8.1855168042470635e-53 < b_2 < 3.634898599408338e+146

    1. Initial program 13.2

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

    3. Applied div-inv13.3

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

    5. Applied associate-*r/13.2

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

    6. Simplified13.2

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

    if 3.634898599408338e+146 < b_2

    1. Initial program 58.5

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

    3. Applied div-inv58.5

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

    4. Taylor expanded around inf 2.0

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

    5. Simplified2.0

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.1855168042470635 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.634898599408338 \cdot 10^{+146}:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))