Average Error: 34.1 → 8.9
Time: 16.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\ \;\;\;\;-\frac{\frac{c \cdot a}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\ \mathbf{elif}\;b_2 \le -5.546621280225112292650318866994441138678 \cdot 10^{-56}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.823335453796603439248590818149160856749 \cdot 10^{131}:\\ \;\;\;\;\frac{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{2 \cdot b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.122942397323538653087473965252285768999 \cdot 10^{137}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -3.408354642852288642375909774932644593067 \cdot 10^{-45}:\\
\;\;\;\;-\frac{\frac{c \cdot a}{a}}{b_2 - \sqrt{b_2 \cdot b_2 - c \cdot a}}\\

\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{-\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{2 \cdot b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r23801 = b_2;
        double r23802 = -r23801;
        double r23803 = r23801 * r23801;
        double r23804 = a;
        double r23805 = c;
        double r23806 = r23804 * r23805;
        double r23807 = r23803 - r23806;
        double r23808 = sqrt(r23807);
        double r23809 = r23802 - r23808;
        double r23810 = r23809 / r23804;
        return r23810;
}
↓
double f(double a, double b_2, double c) {
        double r23811 = b_2;
        double r23812 = -2.1229423973235387e+137;
        bool r23813 = r23811 <= r23812;
        double r23814 = -0.5;
        double r23815 = c;
        double r23816 = r23815 / r23811;
        double r23817 = r23814 * r23816;
        double r23818 = -3.4083546428522886e-45;
        bool r23819 = r23811 <= r23818;
        double r23820 = a;
        double r23821 = r23815 * r23820;
        double r23822 = r23821 / r23820;
        double r23823 = r23811 * r23811;
        double r23824 = r23823 - r23821;
        double r23825 = sqrt(r23824);
        double r23826 = r23811 - r23825;
        double r23827 = r23822 / r23826;
        double r23828 = -r23827;
        double r23829 = -5.546621280225112e-56;
        bool r23830 = r23811 <= r23829;
        double r23831 = 2.8233354537966034e+131;
        bool r23832 = r23811 <= r23831;
        double r23833 = r23825 + r23811;
        double r23834 = -r23833;
        double r23835 = r23834 / r23820;
        double r23836 = 0.5;
        double r23837 = r23811 / r23815;
        double r23838 = r23836 / r23837;
        double r23839 = 2.0;
        double r23840 = r23839 * r23811;
        double r23841 = r23840 / r23820;
        double r23842 = r23838 - r23841;
        double r23843 = r23832 ? r23835 : r23842;
        double r23844 = r23830 ? r23817 : r23843;
        double r23845 = r23819 ? r23828 : r23844;
        double r23846 = r23813 ? r23817 : r23845;
        return r23846;
}

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 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. Simplified61.6

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

    3. 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. Simplified45.1

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

    4. Applied div-inv45.1

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

    6. Applied flip-+45.1

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

    7. Applied distribute-neg-frac45.1

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

    8. Applied associate-*l/45.1

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

    9. Simplified11.6

      \[\leadsto \frac{\color{blue}{\frac{-\left(0 + a \cdot c\right)}{a}}}{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    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. Simplified12.5

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

    4. Applied div-inv12.7

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

    6. Applied pow112.7

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

    7. Applied pow112.7

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

    8. Applied pow-prod-down12.7

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

    9. Simplified12.5

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

    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. Simplified56.4

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

    3. Taylor expanded around inf 2.4

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

    4. Simplified2.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\frac{b_2}{c}} - \frac{b_2 \cdot 2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))