Average Error: 34.3 → 9.8
Time: 10.3s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r51051 = b_2;
        double r51052 = -r51051;
        double r51053 = r51051 * r51051;
        double r51054 = a;
        double r51055 = c;
        double r51056 = r51054 * r51055;
        double r51057 = r51053 - r51056;
        double r51058 = sqrt(r51057);
        double r51059 = r51052 - r51058;
        double r51060 = r51059 / r51054;
        return r51060;
}
↓
double f(double a, double b_2, double c) {
        double r51061 = b_2;
        double r51062 = -2.27187581796005e-81;
        bool r51063 = r51061 <= r51062;
        double r51064 = -0.5;
        double r51065 = c;
        double r51066 = r51065 / r51061;
        double r51067 = r51064 * r51066;
        double r51068 = 3.5836490410280977e+84;
        bool r51069 = r51061 <= r51068;
        double r51070 = -r51061;
        double r51071 = -r51065;
        double r51072 = a;
        double r51073 = r51061 * r51061;
        double r51074 = fma(r51071, r51072, r51073);
        double r51075 = sqrt(r51074);
        double r51076 = r51070 - r51075;
        double r51077 = r51076 / r51072;
        double r51078 = r51070 - r51061;
        double r51079 = r51078 / r51072;
        double r51080 = r51069 ? r51077 : r51079;
        double r51081 = r51063 ? r51067 : r51080;
        return r51081;
}

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 < -2.27187581796005e-81

    1. Initial program 53.4

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

    2. Simplified53.4

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

    3. Taylor expanded around -inf 8.8

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

    if -2.27187581796005e-81 < b_2 < 3.5836490410280977e+84

    1. Initial program 13.1

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

    2. Simplified13.1

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

    3. Taylor expanded around 0 13.1

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

    4. Simplified13.1

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

    if 3.5836490410280977e+84 < b_2

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around 0 3.7

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.271875817960049862049445045905922921165 \cdot 10^{-81}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.583649041028097672662453906912522703022 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{\mathsf{fma}\left(-c, a, b_2 \cdot b_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))