Average Error: 33.8 → 6.5
Time: 15.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.155327002621879240971632151563348379742 \cdot 10^{117}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.622962172940872725615087320648120303101 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 5.179201385701060896596521886473852187744 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.155327002621879240971632151563348379742 \cdot 10^{117}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 3.622962172940872725615087320648120303101 \cdot 10^{-206}:\\
\;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 5.179201385701060896596521886473852187744 \cdot 10^{98}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r63844 = b;
        double r63845 = -r63844;
        double r63846 = r63844 * r63844;
        double r63847 = 4.0;
        double r63848 = a;
        double r63849 = c;
        double r63850 = r63848 * r63849;
        double r63851 = r63847 * r63850;
        double r63852 = r63846 - r63851;
        double r63853 = sqrt(r63852);
        double r63854 = r63845 - r63853;
        double r63855 = 2.0;
        double r63856 = r63855 * r63848;
        double r63857 = r63854 / r63856;
        return r63857;
}
↓
double f(double a, double b, double c) {
        double r63858 = b;
        double r63859 = -9.155327002621879e+117;
        bool r63860 = r63858 <= r63859;
        double r63861 = -1.0;
        double r63862 = c;
        double r63863 = r63862 / r63858;
        double r63864 = r63861 * r63863;
        double r63865 = 3.622962172940873e-206;
        bool r63866 = r63858 <= r63865;
        double r63867 = 2.0;
        double r63868 = r63867 * r63862;
        double r63869 = -r63858;
        double r63870 = r63858 * r63858;
        double r63871 = 4.0;
        double r63872 = a;
        double r63873 = r63872 * r63862;
        double r63874 = r63871 * r63873;
        double r63875 = r63870 - r63874;
        double r63876 = sqrt(r63875);
        double r63877 = r63869 + r63876;
        double r63878 = r63868 / r63877;
        double r63879 = 5.179201385701061e+98;
        bool r63880 = r63858 <= r63879;
        double r63881 = r63869 - r63876;
        double r63882 = r63881 / r63867;
        double r63883 = r63882 / r63872;
        double r63884 = 1.0;
        double r63885 = r63858 / r63872;
        double r63886 = r63863 - r63885;
        double r63887 = r63884 * r63886;
        double r63888 = r63880 ? r63883 : r63887;
        double r63889 = r63866 ? r63878 : r63888;
        double r63890 = r63860 ? r63864 : r63889;
        return r63890;
}

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

Original33.8
Target20.4
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.155327002621879e+117

    1. Initial program 60.8

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

    2. Taylor expanded around -inf 1.8

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

    if -9.155327002621879e+117 < b < 3.622962172940873e-206

    1. Initial program 30.1

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

    3. Applied associate-/r*30.1

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

    5. Applied *-un-lft-identity30.1

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

    6. Applied div-inv30.1

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

    7. Applied times-frac30.1

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

    8. Simplified30.1

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

    10. Applied flip--30.3

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

    11. Applied associate-*l/30.3

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

    12. Simplified14.7

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

    13. Taylor expanded around 0 9.4

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

    if 3.622962172940873e-206 < b < 5.179201385701061e+98

    1. Initial program 8.1

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

    3. Applied associate-/r*8.1

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

    if 5.179201385701061e+98 < b

    1. Initial program 44.9

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.155327002621879240971632151563348379742 \cdot 10^{117}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.622962172940872725615087320648120303101 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 5.179201385701060896596521886473852187744 \cdot 10^{98}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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