Average Error: 34.7 → 10.1
Time: 5.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\
\;\;\;\;\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 r55105 = b;
        double r55106 = -r55105;
        double r55107 = r55105 * r55105;
        double r55108 = 4.0;
        double r55109 = a;
        double r55110 = r55108 * r55109;
        double r55111 = c;
        double r55112 = r55110 * r55111;
        double r55113 = r55107 - r55112;
        double r55114 = sqrt(r55113);
        double r55115 = r55106 + r55114;
        double r55116 = 2.0;
        double r55117 = r55116 * r55109;
        double r55118 = r55115 / r55117;
        return r55118;
}
↓
double f(double a, double b, double c) {
        double r55119 = b;
        double r55120 = -1.2447742914077108e+109;
        bool r55121 = r55119 <= r55120;
        double r55122 = 1.0;
        double r55123 = c;
        double r55124 = r55123 / r55119;
        double r55125 = a;
        double r55126 = r55119 / r55125;
        double r55127 = r55124 - r55126;
        double r55128 = r55122 * r55127;
        double r55129 = 6.485606601696406e-71;
        bool r55130 = r55119 <= r55129;
        double r55131 = -r55119;
        double r55132 = r55119 * r55119;
        double r55133 = 4.0;
        double r55134 = r55133 * r55125;
        double r55135 = r55134 * r55123;
        double r55136 = r55132 - r55135;
        double r55137 = sqrt(r55136);
        double r55138 = r55131 + r55137;
        double r55139 = 1.0;
        double r55140 = 2.0;
        double r55141 = r55140 * r55125;
        double r55142 = r55139 / r55141;
        double r55143 = r55138 * r55142;
        double r55144 = -1.0;
        double r55145 = r55144 * r55124;
        double r55146 = r55130 ? r55143 : r55145;
        double r55147 = r55121 ? r55128 : r55146;
        return r55147;
}

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 < -1.2447742914077108e+109

    1. Initial program 49.3

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

    2. Taylor expanded around -inf 4.0

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

    3. Simplified4.0

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

    if -1.2447742914077108e+109 < b < 6.485606601696406e-71

    1. Initial program 13.5

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

    3. Applied div-inv13.6

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

    if 6.485606601696406e-71 < b

    1. Initial program 53.3

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

    2. Taylor expanded around inf 8.4

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))