Average Error: 34.7 → 10.1
Time: 7.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.244774291407710824026233990502584030865 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 6.485606601696406255086078549712143397431 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 r219864 = b;
        double r219865 = -r219864;
        double r219866 = r219864 * r219864;
        double r219867 = 4.0;
        double r219868 = a;
        double r219869 = r219867 * r219868;
        double r219870 = c;
        double r219871 = r219869 * r219870;
        double r219872 = r219866 - r219871;
        double r219873 = sqrt(r219872);
        double r219874 = r219865 + r219873;
        double r219875 = 2.0;
        double r219876 = r219875 * r219868;
        double r219877 = r219874 / r219876;
        return r219877;
}
↓
double f(double a, double b, double c) {
        double r219878 = b;
        double r219879 = -1.2447742914077108e+109;
        bool r219880 = r219878 <= r219879;
        double r219881 = 1.0;
        double r219882 = c;
        double r219883 = r219882 / r219878;
        double r219884 = a;
        double r219885 = r219878 / r219884;
        double r219886 = r219883 - r219885;
        double r219887 = r219881 * r219886;
        double r219888 = 6.485606601696406e-71;
        bool r219889 = r219878 <= r219888;
        double r219890 = -r219878;
        double r219891 = r219878 * r219878;
        double r219892 = 4.0;
        double r219893 = r219892 * r219884;
        double r219894 = r219893 * r219882;
        double r219895 = r219891 - r219894;
        double r219896 = sqrt(r219895);
        double r219897 = r219890 + r219896;
        double r219898 = 1.0;
        double r219899 = 2.0;
        double r219900 = r219899 * r219884;
        double r219901 = r219898 / r219900;
        double r219902 = r219897 * r219901;
        double r219903 = -1.0;
        double r219904 = r219903 * r219883;
        double r219905 = r219889 ? r219902 : r219904;
        double r219906 = r219880 ? r219887 : r219905;
        return r219906;
}

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

Target

Original34.7
Target21.5
Herbie10.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

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 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))