Average Error: 34.8 → 10.7
Time: 4.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.60514141786167054 \cdot 10^{33}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.60514141786167054 \cdot 10^{33}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\
\;\;\;\;\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 r54042 = b;
        double r54043 = -r54042;
        double r54044 = r54042 * r54042;
        double r54045 = 4.0;
        double r54046 = a;
        double r54047 = r54045 * r54046;
        double r54048 = c;
        double r54049 = r54047 * r54048;
        double r54050 = r54044 - r54049;
        double r54051 = sqrt(r54050);
        double r54052 = r54043 + r54051;
        double r54053 = 2.0;
        double r54054 = r54053 * r54046;
        double r54055 = r54052 / r54054;
        return r54055;
}
↓
double f(double a, double b, double c) {
        double r54056 = b;
        double r54057 = -4.6051414178616705e+33;
        bool r54058 = r54056 <= r54057;
        double r54059 = 1.0;
        double r54060 = c;
        double r54061 = r54060 / r54056;
        double r54062 = a;
        double r54063 = r54056 / r54062;
        double r54064 = r54061 - r54063;
        double r54065 = r54059 * r54064;
        double r54066 = 1.92049775718538e-66;
        bool r54067 = r54056 <= r54066;
        double r54068 = -r54056;
        double r54069 = r54056 * r54056;
        double r54070 = 4.0;
        double r54071 = r54070 * r54062;
        double r54072 = r54071 * r54060;
        double r54073 = r54069 - r54072;
        double r54074 = sqrt(r54073);
        double r54075 = r54068 + r54074;
        double r54076 = 1.0;
        double r54077 = 2.0;
        double r54078 = r54077 * r54062;
        double r54079 = r54076 / r54078;
        double r54080 = r54075 * r54079;
        double r54081 = -1.0;
        double r54082 = r54081 * r54061;
        double r54083 = r54067 ? r54080 : r54082;
        double r54084 = r54058 ? r54065 : r54083;
        return r54084;
}

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 < -4.6051414178616705e+33

    1. Initial program 36.3

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

    2. Taylor expanded around -inf 6.9

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

    3. Simplified6.9

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

    if -4.6051414178616705e+33 < b < 1.92049775718538e-66

    1. Initial program 15.2

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

    3. Applied div-inv15.3

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

    if 1.92049775718538e-66 < b

    1. Initial program 54.0

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

    2. Taylor expanded around inf 8.1

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.60514141786167054 \cdot 10^{33}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.92049775718538 \cdot 10^{-66}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))