Average Error: 33.1 → 10.3
Time: 21.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7194502457644758 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.7194502457644758 \cdot 10^{+64}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\
\;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1102143 = b;
        double r1102144 = -r1102143;
        double r1102145 = r1102143 * r1102143;
        double r1102146 = 4.0;
        double r1102147 = a;
        double r1102148 = r1102146 * r1102147;
        double r1102149 = c;
        double r1102150 = r1102148 * r1102149;
        double r1102151 = r1102145 - r1102150;
        double r1102152 = sqrt(r1102151);
        double r1102153 = r1102144 + r1102152;
        double r1102154 = 2.0;
        double r1102155 = r1102154 * r1102147;
        double r1102156 = r1102153 / r1102155;
        return r1102156;
}
↓
double f(double a, double b, double c) {
        double r1102157 = b;
        double r1102158 = -1.7194502457644758e+64;
        bool r1102159 = r1102157 <= r1102158;
        double r1102160 = c;
        double r1102161 = r1102160 / r1102157;
        double r1102162 = a;
        double r1102163 = r1102157 / r1102162;
        double r1102164 = r1102161 - r1102163;
        double r1102165 = 2.0;
        double r1102166 = r1102164 * r1102165;
        double r1102167 = r1102166 / r1102165;
        double r1102168 = 9.831724396970673e-110;
        bool r1102169 = r1102157 <= r1102168;
        double r1102170 = 1.0;
        double r1102171 = r1102170 / r1102162;
        double r1102172 = r1102157 * r1102157;
        double r1102173 = r1102162 * r1102160;
        double r1102174 = 4.0;
        double r1102175 = r1102173 * r1102174;
        double r1102176 = r1102172 - r1102175;
        double r1102177 = sqrt(r1102176);
        double r1102178 = r1102177 - r1102157;
        double r1102179 = r1102171 * r1102178;
        double r1102180 = r1102179 / r1102165;
        double r1102181 = -2.0;
        double r1102182 = r1102181 * r1102161;
        double r1102183 = r1102182 / r1102165;
        double r1102184 = r1102169 ? r1102180 : r1102183;
        double r1102185 = r1102159 ? r1102167 : r1102184;
        return r1102185;
}

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 < -1.7194502457644758e+64

    1. Initial program 37.7

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

    2. Simplified37.7

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

    3. Taylor expanded around -inf 5.2

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

    4. Simplified5.2

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

    if -1.7194502457644758e+64 < b < 9.831724396970673e-110

    1. Initial program 12.1

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

    2. Simplified12.1

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

    4. Applied div-inv12.2

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

    6. Applied *-commutative12.2

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

    if 9.831724396970673e-110 < b

    1. Initial program 51.0

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

    2. Simplified51.0

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

    4. Applied div-inv51.0

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

    5. Taylor expanded around inf 10.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7194502457644758 \cdot 10^{+64}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 9.831724396970673 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))