Average Error: 33.3 → 10.4
Time: 21.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.961198324014865 \cdot 10^{-88}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 6.384705165981893 \cdot 10^{+101}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r641040 = b_2;
        double r641041 = -r641040;
        double r641042 = r641040 * r641040;
        double r641043 = a;
        double r641044 = c;
        double r641045 = r641043 * r641044;
        double r641046 = r641042 - r641045;
        double r641047 = sqrt(r641046);
        double r641048 = r641041 - r641047;
        double r641049 = r641048 / r641043;
        return r641049;
}
↓
double f(double a, double b_2, double c) {
        double r641050 = b_2;
        double r641051 = -5.961198324014865e-88;
        bool r641052 = r641050 <= r641051;
        double r641053 = -0.5;
        double r641054 = c;
        double r641055 = r641054 / r641050;
        double r641056 = r641053 * r641055;
        double r641057 = 6.384705165981893e+101;
        bool r641058 = r641050 <= r641057;
        double r641059 = -r641050;
        double r641060 = a;
        double r641061 = r641059 / r641060;
        double r641062 = r641050 * r641050;
        double r641063 = r641054 * r641060;
        double r641064 = r641062 - r641063;
        double r641065 = sqrt(r641064);
        double r641066 = r641065 / r641060;
        double r641067 = r641061 - r641066;
        double r641068 = r641050 / r641060;
        double r641069 = -2.0;
        double r641070 = 0.5;
        double r641071 = r641050 / r641054;
        double r641072 = r641070 / r641071;
        double r641073 = fma(r641068, r641069, r641072);
        double r641074 = r641058 ? r641067 : r641073;
        double r641075 = r641052 ? r641056 : r641074;
        return r641075;
}

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 < -5.961198324014865e-88

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 9.8

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

    if -5.961198324014865e-88 < b_2 < 6.384705165981893e+101

    1. Initial program 13.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    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 un-div-inv13.1

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

    7. Applied div-sub13.1

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

    if 6.384705165981893e+101 < b_2

    1. Initial program 43.8

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

    2. Taylor expanded around inf 3.7

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

    3. Simplified3.7

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{\frac{1}{2}}{\frac{b_2}{c}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))