Average Error: 34.6 → 6.3
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\
\;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\

\mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r85862 = b;
        double r85863 = -r85862;
        double r85864 = r85862 * r85862;
        double r85865 = 4.0;
        double r85866 = a;
        double r85867 = c;
        double r85868 = r85866 * r85867;
        double r85869 = r85865 * r85868;
        double r85870 = r85864 - r85869;
        double r85871 = sqrt(r85870);
        double r85872 = r85863 - r85871;
        double r85873 = 2.0;
        double r85874 = r85873 * r85866;
        double r85875 = r85872 / r85874;
        return r85875;
}
↓
double f(double a, double b, double c) {
        double r85876 = b;
        double r85877 = -5.263290697710818e+146;
        bool r85878 = r85876 <= r85877;
        double r85879 = -1.0;
        double r85880 = c;
        double r85881 = r85880 / r85876;
        double r85882 = r85879 * r85881;
        double r85883 = -2.182382645844659e-295;
        bool r85884 = r85876 <= r85883;
        double r85885 = 1.0;
        double r85886 = 2.0;
        double r85887 = a;
        double r85888 = r85886 * r85887;
        double r85889 = 4.0;
        double r85890 = r85889 * r85887;
        double r85891 = r85888 / r85890;
        double r85892 = r85876 * r85876;
        double r85893 = r85887 * r85880;
        double r85894 = r85889 * r85893;
        double r85895 = r85892 - r85894;
        double r85896 = sqrt(r85895);
        double r85897 = r85896 - r85876;
        double r85898 = r85880 / r85897;
        double r85899 = r85891 / r85898;
        double r85900 = r85885 / r85899;
        double r85901 = 3.1607591925776442e+143;
        bool r85902 = r85876 <= r85901;
        double r85903 = -r85876;
        double r85904 = r85903 - r85896;
        double r85905 = r85904 / r85888;
        double r85906 = 1.0;
        double r85907 = r85876 / r85887;
        double r85908 = r85881 - r85907;
        double r85909 = r85906 * r85908;
        double r85910 = r85902 ? r85905 : r85909;
        double r85911 = r85884 ? r85900 : r85910;
        double r85912 = r85878 ? r85882 : r85911;
        return r85912;
}

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

Original34.6
Target20.9
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.263290697710818e+146

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 1.3

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

    if -5.263290697710818e+146 < b < -2.182382645844659e-295

    1. Initial program 34.7

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

    3. Applied flip--34.7

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

    4. Simplified15.7

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

    5. Simplified15.7

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

    7. Applied *-un-lft-identity15.7

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

    8. Applied *-un-lft-identity15.7

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

    9. Applied times-frac15.7

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

    10. Applied associate-/l*15.9

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

    11. Simplified15.9

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

    13. Applied *-un-lft-identity15.9

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

    14. Applied times-frac13.5

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

    15. Applied associate-/r*7.6

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

    16. Simplified7.6

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

    if -2.182382645844659e-295 < b < 3.1607591925776442e+143

    1. Initial program 9.3

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

    if 3.1607591925776442e+143 < b

    1. Initial program 59.6

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

    2. Taylor expanded around inf 2.3

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

    3. Simplified2.3

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.263290697710817942239037357803149075237 \cdot 10^{146}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.182382645844658784648715405900710208288 \cdot 10^{-295}:\\ \;\;\;\;\frac{1}{\frac{\frac{2 \cdot a}{4 \cdot a}}{\frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.160759192577644243019157975166466824718 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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)))