The quadratic formula (r1)

Average Error: 33.5 → 11.9

Time: 45.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a} \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3098862 = b;
        double r3098863 = -r3098862;
        double r3098864 = r3098862 * r3098862;
        double r3098865 = 4.0;
        double r3098866 = a;
        double r3098867 = r3098865 * r3098866;
        double r3098868 = c;
        double r3098869 = r3098867 * r3098868;
        double r3098870 = r3098864 - r3098869;
        double r3098871 = sqrt(r3098870);
        double r3098872 = r3098863 + r3098871;
        double r3098873 = 2.0;
        double r3098874 = r3098873 * r3098866;
        double r3098875 = r3098872 / r3098874;
        return r3098875;
}

↓

double f(double a, double b, double c) {
        double r3098876 = b;
        double r3098877 = -1.3725796156555912e+127;
        bool r3098878 = r3098876 <= r3098877;
        double r3098879 = c;
        double r3098880 = r3098879 / r3098876;
        double r3098881 = a;
        double r3098882 = r3098876 / r3098881;
        double r3098883 = r3098880 - r3098882;
        double r3098884 = 2.0;
        double r3098885 = r3098883 * r3098884;
        double r3098886 = r3098885 / r3098884;
        double r3098887 = 3.207624111695675e-187;
        bool r3098888 = r3098876 <= r3098887;
        double r3098889 = -4.0;
        double r3098890 = r3098881 * r3098889;
        double r3098891 = r3098890 * r3098879;
        double r3098892 = fma(r3098876, r3098876, r3098891);
        double r3098893 = sqrt(r3098892);
        double r3098894 = r3098893 / r3098881;
        double r3098895 = r3098894 - r3098882;
        double r3098896 = r3098895 / r3098884;
        double r3098897 = 4.664677641347216e-111;
        bool r3098898 = r3098876 <= r3098897;
        double r3098899 = -2.0;
        double r3098900 = r3098899 * r3098880;
        double r3098901 = r3098900 / r3098884;
        double r3098902 = 1.922674299151799e-16;
        bool r3098903 = r3098876 <= r3098902;
        double r3098904 = r3098876 * r3098876;
        double r3098905 = fma(r3098879, r3098890, r3098904);
        double r3098906 = r3098905 - r3098904;
        double r3098907 = r3098906 / r3098881;
        double r3098908 = 1.0;
        double r3098909 = sqrt(r3098905);
        double r3098910 = r3098876 + r3098909;
        double r3098911 = r3098908 / r3098910;
        double r3098912 = r3098907 * r3098911;
        double r3098913 = r3098912 / r3098884;
        double r3098914 = r3098903 ? r3098913 : r3098901;
        double r3098915 = r3098898 ? r3098901 : r3098914;
        double r3098916 = r3098888 ? r3098896 : r3098915;
        double r3098917 = r3098878 ? r3098886 : r3098916;
        return r3098917;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.5
Target	21.2
Herbie	11.9

\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -1.3725796156555912e+127

   1. Initial program 51.4

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

   2. Simplified 51.4

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

   3. Taylor expanded around -inf 2.3

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

   4. Simplified 2.3

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

   if -1.3725796156555912e+127 < b < 3.207624111695675e-187

   1. Initial program 10.4

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

   2. Simplified 10.4

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

   4. Applied div-sub 10.4

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

   if 3.207624111695675e-187 < b < 4.664677641347216e-111 or 1.922674299151799e-16 < b

   1. Initial program 50.1

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

   2. Simplified 50.1

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

   3. Taylor expanded around inf 12.0

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

   if 4.664677641347216e-111 < b < 1.922674299151799e-16

   1. Initial program 36.3

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

   2. Simplified 36.3

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

   4. Applied *-un-lft-identity 36.3

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

   5. Applied *-un-lft-identity 36.3

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

   6. Applied distribute-lft-out-- 36.3

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

   7. Applied associate-/l* 36.4

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

   9. Applied flip-- 36.4

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

   10. Applied associate-/r/ 36.5

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

   11. Applied add-cube-cbrt 36.5

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

   12. Applied times-frac 36.5

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

   13. Simplified 36.4

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

   14. Simplified 36.4

       \[\leadsto \frac{\frac{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right) - b \cdot b}{a} \cdot \color{blue}{\frac{1}{b + \sqrt{\mathsf{fma}\left(c, -4 \cdot a, b \cdot b\right)}}}}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3725796156555912 \cdot 10^{+127}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.207624111695675 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(a \cdot -4\right) \cdot c\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{elif}\;b \le 4.664677641347216 \cdot 10^{-111}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.922674299151799 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right) - b \cdot b}{a} \cdot \frac{1}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 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)))