Average Error: 34.1 → 8.9
Time: 20.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\
\;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\

\mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\
\;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r862801 = b_2;
        double r862802 = -r862801;
        double r862803 = r862801 * r862801;
        double r862804 = a;
        double r862805 = c;
        double r862806 = r862804 * r862805;
        double r862807 = r862803 - r862806;
        double r862808 = sqrt(r862807);
        double r862809 = r862802 - r862808;
        double r862810 = r862809 / r862804;
        return r862810;
}
↓
double f(double a, double b_2, double c) {
        double r862811 = b_2;
        double r862812 = -2.1229423973235387e+137;
        bool r862813 = r862811 <= r862812;
        double r862814 = -0.5;
        double r862815 = c;
        double r862816 = r862815 / r862811;
        double r862817 = r862814 * r862816;
        double r862818 = -3.4083546428522886e-45;
        bool r862819 = r862811 <= r862818;
        double r862820 = a;
        double r862821 = r862820 * r862815;
        double r862822 = r862811 * r862811;
        double r862823 = r862822 - r862822;
        double r862824 = r862821 + r862823;
        double r862825 = r862824 / r862820;
        double r862826 = r862822 - r862821;
        double r862827 = sqrt(r862826);
        double r862828 = -r862811;
        double r862829 = r862827 + r862828;
        double r862830 = r862825 / r862829;
        double r862831 = -5.546621280225112e-56;
        bool r862832 = r862811 <= r862831;
        double r862833 = 2.8233354537966034e+131;
        bool r862834 = r862811 <= r862833;
        double r862835 = r862827 + r862811;
        double r862836 = r862835 / r862820;
        double r862837 = -r862836;
        double r862838 = r862811 / r862820;
        double r862839 = -2.0;
        double r862840 = 0.5;
        double r862841 = r862816 * r862840;
        double r862842 = fma(r862838, r862839, r862841);
        double r862843 = r862834 ? r862837 : r862842;
        double r862844 = r862832 ? r862817 : r862843;
        double r862845 = r862819 ? r862830 : r862844;
        double r862846 = r862813 ? r862817 : r862845;
        return r862846;
}

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 < -2.1229423973235387e+137 or -3.4083546428522886e-45 < b_2 < -5.546621280225112e-56

    1. Initial program 61.6

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

    2. Taylor expanded around -inf 2.3

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

    if -2.1229423973235387e+137 < b_2 < -3.4083546428522886e-45

    1. Initial program 45.1

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

    3. Applied div-inv45.1

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

    5. Applied flip--45.1

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

    6. Applied associate-*l/45.1

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

    7. Simplified11.6

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

    if -5.546621280225112e-56 < b_2 < 2.8233354537966034e+131

    1. Initial program 12.5

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

    3. Applied div-inv12.7

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

    5. Applied associate-*r/12.5

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

    6. Simplified12.5

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

    if 2.8233354537966034e+131 < b_2

    1. Initial program 56.4

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

    3. Applied div-inv56.4

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

    4. Taylor expanded around inf 2.4

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

    5. Simplified2.4

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{a \cdot c + \left(b_2 \cdot b_2 - b_2 \cdot b_2\right)}{a}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}\\ \mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\ \;\;\;\;-\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{c}{b_2} \cdot \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))