Average Error: 33.1 → 9.2
Time: 1.2m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.754487595174753 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.873738165909194 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot \left(\left(c \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}\right)}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -1.6124744946043857 \cdot 10^{+143}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.754487595174753 \cdot 10^{-113}:\\
\;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 9.873738165909194 \cdot 10^{+16}:\\
\;\;\;\;\frac{a \cdot \left(\left(c \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}\right)}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6771026 = b;
        double r6771027 = -r6771026;
        double r6771028 = r6771026 * r6771026;
        double r6771029 = 4.0;
        double r6771030 = a;
        double r6771031 = r6771029 * r6771030;
        double r6771032 = c;
        double r6771033 = r6771031 * r6771032;
        double r6771034 = r6771028 - r6771033;
        double r6771035 = sqrt(r6771034);
        double r6771036 = r6771027 + r6771035;
        double r6771037 = 2.0;
        double r6771038 = r6771037 * r6771030;
        double r6771039 = r6771036 / r6771038;
        return r6771039;
}
↓
double f(double a, double b, double c) {
        double r6771040 = b;
        double r6771041 = -1.6124744946043857e+143;
        bool r6771042 = r6771040 <= r6771041;
        double r6771043 = c;
        double r6771044 = r6771043 / r6771040;
        double r6771045 = a;
        double r6771046 = r6771040 / r6771045;
        double r6771047 = r6771044 - r6771046;
        double r6771048 = 1.754487595174753e-113;
        bool r6771049 = r6771040 <= r6771048;
        double r6771050 = -4.0;
        double r6771051 = r6771043 * r6771050;
        double r6771052 = r6771051 * r6771045;
        double r6771053 = r6771040 * r6771040;
        double r6771054 = r6771052 + r6771053;
        double r6771055 = sqrt(r6771054);
        double r6771056 = r6771055 - r6771040;
        double r6771057 = 2.0;
        double r6771058 = r6771045 * r6771057;
        double r6771059 = r6771056 / r6771058;
        double r6771060 = 9.873738165909194e+16;
        bool r6771061 = r6771040 <= r6771060;
        double r6771062 = 0.5;
        double r6771063 = r6771062 / r6771045;
        double r6771064 = r6771051 * r6771063;
        double r6771065 = r6771045 * r6771064;
        double r6771066 = r6771055 + r6771040;
        double r6771067 = r6771065 / r6771066;
        double r6771068 = -r6771044;
        double r6771069 = r6771061 ? r6771067 : r6771068;
        double r6771070 = r6771049 ? r6771059 : r6771069;
        double r6771071 = r6771042 ? r6771047 : r6771070;
        return r6771071;
}

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 < -1.6124744946043857e+143

    1. Initial program 57.1

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

    2. Simplified57.1

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

    3. Taylor expanded around -inf 57.1

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

    4. Simplified57.1

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

    5. Taylor expanded around -inf 2.6

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

    if -1.6124744946043857e+143 < b < 1.754487595174753e-113

    1. Initial program 11.1

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

    2. Simplified11.1

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

    3. Taylor expanded around -inf 11.1

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

    4. Simplified11.1

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

    if 1.754487595174753e-113 < b < 9.873738165909194e+16

    1. Initial program 38.5

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

    2. Simplified38.5

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

    3. Taylor expanded around -inf 38.5

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

    4. Simplified38.5

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

    6. Applied clear-num38.5

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

    8. Applied flip--38.6

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

    9. Applied associate-/r/38.6

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

    10. Applied associate-/r*38.6

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

    11. Simplified18.7

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

    if 9.873738165909194e+16 < b

    1. Initial program 54.7

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

    2. Simplified54.7

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

    3. Taylor expanded around inf 5.7

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

    4. Simplified5.7

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.754487595174753 \cdot 10^{-113}:\\ \;\;\;\;\frac{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 9.873738165909194 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot \left(\left(c \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}\right)}{\sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))