Average Error: 34.7 → 10.8
Time: 5.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r65110 = b;
        double r65111 = -r65110;
        double r65112 = r65110 * r65110;
        double r65113 = 4.0;
        double r65114 = a;
        double r65115 = r65113 * r65114;
        double r65116 = c;
        double r65117 = r65115 * r65116;
        double r65118 = r65112 - r65117;
        double r65119 = sqrt(r65118);
        double r65120 = r65111 + r65119;
        double r65121 = 2.0;
        double r65122 = r65121 * r65114;
        double r65123 = r65120 / r65122;
        return r65123;
}
↓
double f(double a, double b, double c) {
        double r65124 = b;
        double r65125 = -6.371698442415157e+150;
        bool r65126 = r65124 <= r65125;
        double r65127 = 1.0;
        double r65128 = c;
        double r65129 = r65128 / r65124;
        double r65130 = a;
        double r65131 = r65124 / r65130;
        double r65132 = r65129 - r65131;
        double r65133 = r65127 * r65132;
        double r65134 = 2.3065444773801163e-129;
        bool r65135 = r65124 <= r65134;
        double r65136 = -r65124;
        double r65137 = r65124 * r65124;
        double r65138 = 4.0;
        double r65139 = r65138 * r65130;
        double r65140 = r65139 * r65128;
        double r65141 = r65137 - r65140;
        double r65142 = sqrt(r65141);
        double r65143 = r65136 + r65142;
        double r65144 = 1.0;
        double r65145 = 2.0;
        double r65146 = r65145 * r65130;
        double r65147 = r65144 / r65146;
        double r65148 = r65143 * r65147;
        double r65149 = -1.0;
        double r65150 = r65149 * r65129;
        double r65151 = r65135 ? r65148 : r65150;
        double r65152 = r65126 ? r65133 : r65151;
        return r65152;
}

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 3 regimes

  2. if b < -6.371698442415157e+150

    1. Initial program 63.0

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

    2. Taylor expanded around -inf 2.5

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

    3. Simplified2.5

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

    if -6.371698442415157e+150 < b < 2.3065444773801163e-129

    1. Initial program 11.3

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

    3. Applied div-inv11.5

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

    if 2.3065444773801163e-129 < b

    1. Initial program 51.5

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

    2. Taylor expanded around inf 12.3

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.371698442415157100029538982618411822116 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.306544477380116301747543706493703838768 \cdot 10^{-129}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))