Average Error: 34.3 → 6.2
Time: 5.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.00336887744975224 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.72191682831812008 \cdot 10^{-246}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.54044500863652378 \cdot 10^{145}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -8.00336887744975224 \cdot 10^{118}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.72191682831812008 \cdot 10^{-246}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 5.54044500863652378 \cdot 10^{145}:\\
\;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r61042 = b;
        double r61043 = -r61042;
        double r61044 = r61042 * r61042;
        double r61045 = 4.0;
        double r61046 = a;
        double r61047 = r61045 * r61046;
        double r61048 = c;
        double r61049 = r61047 * r61048;
        double r61050 = r61044 - r61049;
        double r61051 = sqrt(r61050);
        double r61052 = r61043 + r61051;
        double r61053 = 2.0;
        double r61054 = r61053 * r61046;
        double r61055 = r61052 / r61054;
        return r61055;
}
↓
double f(double a, double b, double c) {
        double r61056 = b;
        double r61057 = -8.003368877449752e+118;
        bool r61058 = r61056 <= r61057;
        double r61059 = 1.0;
        double r61060 = c;
        double r61061 = r61060 / r61056;
        double r61062 = a;
        double r61063 = r61056 / r61062;
        double r61064 = r61061 - r61063;
        double r61065 = r61059 * r61064;
        double r61066 = -3.72191682831812e-246;
        bool r61067 = r61056 <= r61066;
        double r61068 = -r61056;
        double r61069 = r61056 * r61056;
        double r61070 = 4.0;
        double r61071 = r61070 * r61062;
        double r61072 = r61071 * r61060;
        double r61073 = r61069 - r61072;
        double r61074 = sqrt(r61073);
        double r61075 = r61068 + r61074;
        double r61076 = 2.0;
        double r61077 = r61076 * r61062;
        double r61078 = r61075 / r61077;
        double r61079 = 5.540445008636524e+145;
        bool r61080 = r61056 <= r61079;
        double r61081 = 1.0;
        double r61082 = r61076 / r61070;
        double r61083 = r61081 / r61082;
        double r61084 = r61083 * r61060;
        double r61085 = r61068 - r61074;
        double r61086 = r61084 / r61085;
        double r61087 = -1.0;
        double r61088 = r61087 * r61061;
        double r61089 = r61080 ? r61086 : r61088;
        double r61090 = r61067 ? r61078 : r61089;
        double r61091 = r61058 ? r61065 : r61090;
        return r61091;
}

Error

Bits error versus a

Bits error versus b

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 < -8.003368877449752e+118

    1. Initial program 52.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around -inf 2.9

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

    3. Simplified2.9

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

    if -8.003368877449752e+118 < b < -3.72191682831812e-246

    1. Initial program 7.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    if -3.72191682831812e-246 < b < 5.540445008636524e+145

    1. Initial program 32.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+32.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num16.0

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

    7. Simplified14.7

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{4 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
    8. Using strategy rm

    9. Applied times-frac14.7

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{2}{4} \cdot \frac{a}{a \cdot c}\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]

    10. Simplified9.1

      \[\leadsto \frac{1}{\left(\frac{2}{4} \cdot \color{blue}{\frac{1}{c}}\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\]
    11. Using strategy rm

    12. Applied associate-/r*8.6

      \[\leadsto \color{blue}{\frac{\frac{1}{\frac{2}{4} \cdot \frac{1}{c}}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

    13. Simplified8.6

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{2}{4}} \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\]

    if 5.540445008636524e+145 < b

    1. Initial program 63.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

    2. Taylor expanded around inf 1.8

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

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.00336887744975224 \cdot 10^{118}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.72191682831812008 \cdot 10^{-246}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.54044500863652378 \cdot 10^{145}:\\ \;\;\;\;\frac{\frac{1}{\frac{2}{4}} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))