Average Error: 34.3 → 9.2
Time: 4.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.850740894150185 \cdot 10^{20}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.1806208921043762 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.29361288009581358 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.850740894150185 \cdot 10^{20}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -6.1806208921043762 \cdot 10^{-95}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le -2.125553485370055 \cdot 10^{-113}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.29361288009581358 \cdot 10^{112}:\\
\;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r104865 = b;
        double r104866 = -r104865;
        double r104867 = r104865 * r104865;
        double r104868 = 4.0;
        double r104869 = a;
        double r104870 = c;
        double r104871 = r104869 * r104870;
        double r104872 = r104868 * r104871;
        double r104873 = r104867 - r104872;
        double r104874 = sqrt(r104873);
        double r104875 = r104866 - r104874;
        double r104876 = 2.0;
        double r104877 = r104876 * r104869;
        double r104878 = r104875 / r104877;
        return r104878;
}
↓
double f(double a, double b, double c) {
        double r104879 = b;
        double r104880 = -1.8507408941501855e+20;
        bool r104881 = r104879 <= r104880;
        double r104882 = -1.0;
        double r104883 = c;
        double r104884 = r104883 / r104879;
        double r104885 = r104882 * r104884;
        double r104886 = -6.180620892104376e-95;
        bool r104887 = r104879 <= r104886;
        double r104888 = 1.0;
        double r104889 = 2.0;
        double r104890 = r104888 / r104889;
        double r104891 = 2.0;
        double r104892 = pow(r104879, r104891);
        double r104893 = r104892 - r104892;
        double r104894 = 4.0;
        double r104895 = a;
        double r104896 = r104895 * r104883;
        double r104897 = r104894 * r104896;
        double r104898 = r104893 + r104897;
        double r104899 = r104890 * r104898;
        double r104900 = r104899 / r104895;
        double r104901 = -r104879;
        double r104902 = r104879 * r104879;
        double r104903 = r104902 - r104897;
        double r104904 = sqrt(r104903);
        double r104905 = r104901 + r104904;
        double r104906 = r104888 / r104905;
        double r104907 = r104900 * r104906;
        double r104908 = -2.125553485370055e-113;
        bool r104909 = r104879 <= r104908;
        double r104910 = 6.293612880095814e+112;
        bool r104911 = r104879 <= r104910;
        double r104912 = r104889 * r104895;
        double r104913 = r104888 / r104912;
        double r104914 = r104901 - r104904;
        double r104915 = r104913 * r104914;
        double r104916 = r104879 / r104895;
        double r104917 = r104882 * r104916;
        double r104918 = r104911 ? r104915 : r104917;
        double r104919 = r104909 ? r104885 : r104918;
        double r104920 = r104887 ? r104907 : r104919;
        double r104921 = r104881 ? r104885 : r104920;
        return r104921;
}

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

Original34.3
Target21.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8507408941501855e+20 or -6.180620892104376e-95 < b < -2.125553485370055e-113

    1. Initial program 54.9

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

    2. Taylor expanded around -inf 6.3

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

    if -1.8507408941501855e+20 < b < -6.180620892104376e-95

    1. Initial program 39.1

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

    3. Applied clear-num39.1

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

    5. Applied flip--39.1

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

    6. Applied associate-/r/39.2

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

    7. Applied add-cube-cbrt39.2

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{2 \cdot a}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\]

    8. Applied times-frac39.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{\sqrt[3]{1}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified14.4

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}} \cdot \frac{\sqrt[3]{1}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

    10. Simplified14.4

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

    if -2.125553485370055e-113 < b < 6.293612880095814e+112

    1. Initial program 12.2

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

    3. Applied clear-num12.3

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

    5. Applied div-inv12.4

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

    6. Applied add-cube-cbrt12.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(2 \cdot a\right) \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    7. Applied times-frac12.4

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{2 \cdot a} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    8. Simplified12.4

      \[\leadsto \color{blue}{\frac{1}{2 \cdot a}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified12.3

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

    if 6.293612880095814e+112 < b

    1. Initial program 49.8

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

    3. Applied clear-num49.8

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

    4. Taylor expanded around 0 2.9

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.850740894150185 \cdot 10^{20}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.1806208921043762 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a} \cdot \frac{1}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le -2.125553485370055 \cdot 10^{-113}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.29361288009581358 \cdot 10^{112}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(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)))