Average Error: 32.6 → 9.7
Time: 37.2s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.90131991727783 \cdot 10^{-39}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 4.012768074517757 \cdot 10^{+87}:\\
\;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\

\mathbf{else}:\\
\;\;\;\;-2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r896806 = b_2;
        double r896807 = -r896806;
        double r896808 = r896806 * r896806;
        double r896809 = a;
        double r896810 = c;
        double r896811 = r896809 * r896810;
        double r896812 = r896808 - r896811;
        double r896813 = sqrt(r896812);
        double r896814 = r896807 - r896813;
        double r896815 = r896814 / r896809;
        return r896815;
}
↓
double f(double a, double b_2, double c) {
        double r896816 = b_2;
        double r896817 = -6.90131991727783e-39;
        bool r896818 = r896816 <= r896817;
        double r896819 = -0.5;
        double r896820 = c;
        double r896821 = r896820 / r896816;
        double r896822 = r896819 * r896821;
        double r896823 = 4.012768074517757e+87;
        bool r896824 = r896816 <= r896823;
        double r896825 = 1.0;
        double r896826 = a;
        double r896827 = r896825 / r896826;
        double r896828 = -r896816;
        double r896829 = r896816 * r896816;
        double r896830 = r896820 * r896826;
        double r896831 = r896829 - r896830;
        double r896832 = sqrt(r896831);
        double r896833 = r896828 - r896832;
        double r896834 = r896825 / r896833;
        double r896835 = r896827 / r896834;
        double r896836 = -2.0;
        double r896837 = r896816 / r896826;
        double r896838 = r896836 * r896837;
        double r896839 = r896824 ? r896835 : r896838;
        double r896840 = r896818 ? r896822 : r896839;
        return r896840;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -6.90131991727783e-39

    1. Initial program 53.0

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

    2. Taylor expanded around -inf 7.9

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

    if -6.90131991727783e-39 < b_2 < 4.012768074517757e+87

    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 *-un-lft-identity13.2

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

    4. Applied associate-/l*13.3

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

    6. Applied div-inv13.4

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

    7. Applied associate-/r*13.4

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

    if 4.012768074517757e+87 < b_2

    1. Initial program 41.3

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

    3. Applied *-un-lft-identity41.3

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

    4. Applied associate-/l*41.4

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

    5. Taylor expanded around 0 3.3

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.90131991727783 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 4.012768074517757 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{1}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +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))