Average Error: 34.2 → 8.9
Time: 14.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.50529903241060843 \cdot 10^{27}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -6.63395785424786023 \cdot 10^{-258}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.16896907782470713 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}{a}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.50529903241060843 \cdot 10^{27}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -6.63395785424786023 \cdot 10^{-258}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.16896907782470713 \cdot 10^{-19}:\\
\;\;\;\;\frac{\frac{1}{\frac{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}{a}}{c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r190828 = b;
        double r190829 = -r190828;
        double r190830 = r190828 * r190828;
        double r190831 = 4.0;
        double r190832 = a;
        double r190833 = r190831 * r190832;
        double r190834 = c;
        double r190835 = r190833 * r190834;
        double r190836 = r190830 - r190835;
        double r190837 = sqrt(r190836);
        double r190838 = r190829 + r190837;
        double r190839 = 2.0;
        double r190840 = r190839 * r190832;
        double r190841 = r190838 / r190840;
        return r190841;
}
↓
double f(double a, double b, double c) {
        double r190842 = b;
        double r190843 = -1.5052990324106084e+27;
        bool r190844 = r190842 <= r190843;
        double r190845 = 1.0;
        double r190846 = c;
        double r190847 = r190846 / r190842;
        double r190848 = a;
        double r190849 = r190842 / r190848;
        double r190850 = r190847 - r190849;
        double r190851 = r190845 * r190850;
        double r190852 = -6.63395785424786e-258;
        bool r190853 = r190842 <= r190852;
        double r190854 = -r190842;
        double r190855 = r190842 * r190842;
        double r190856 = 4.0;
        double r190857 = r190856 * r190848;
        double r190858 = r190857 * r190846;
        double r190859 = r190855 - r190858;
        double r190860 = sqrt(r190859);
        double r190861 = r190854 + r190860;
        double r190862 = 1.0;
        double r190863 = 2.0;
        double r190864 = r190863 * r190848;
        double r190865 = r190862 / r190864;
        double r190866 = r190861 * r190865;
        double r190867 = 1.1689690778247071e-19;
        bool r190868 = r190842 <= r190867;
        double r190869 = r190854 - r190860;
        double r190870 = r190869 / r190856;
        double r190871 = r190870 / r190848;
        double r190872 = r190871 / r190846;
        double r190873 = r190862 / r190872;
        double r190874 = r190873 / r190864;
        double r190875 = -1.0;
        double r190876 = r190875 * r190847;
        double r190877 = r190868 ? r190874 : r190876;
        double r190878 = r190853 ? r190866 : r190877;
        double r190879 = r190844 ? r190851 : r190878;
        return r190879;
}

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

Original34.2
Target21.2
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.5052990324106084e+27

    1. Initial program 35.7

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

    2. Taylor expanded around -inf 6.7

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

    3. Simplified6.7

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

    if -1.5052990324106084e+27 < b < -6.63395785424786e-258

    1. Initial program 9.0

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

    3. Applied div-inv9.1

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

    if -6.63395785424786e-258 < b < 1.1689690778247071e-19

    1. Initial program 23.8

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

    3. Applied flip-+23.9

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

    4. Simplified17.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied clear-num17.8

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{0 + 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]

    7. Simplified17.8

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

    9. Applied associate-/r*15.2

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

    if 1.1689690778247071e-19 < b

    1. Initial program 55.0

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

    2. Taylor expanded around inf 6.2

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.50529903241060843 \cdot 10^{27}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -6.63395785424786023 \cdot 10^{-258}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.16896907782470713 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{4}}{a}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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)))