The quadratic formula (r1)

Average Error: 34.7 → 6.9

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \mathbf{elif}\;b \le -1.5018009284516093 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.0268171554574348 \cdot 10^{125}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r89841 = b;
        double r89842 = -r89841;
        double r89843 = r89841 * r89841;
        double r89844 = 4.0;
        double r89845 = a;
        double r89846 = r89844 * r89845;
        double r89847 = c;
        double r89848 = r89846 * r89847;
        double r89849 = r89843 - r89848;
        double r89850 = sqrt(r89849);
        double r89851 = r89842 + r89850;
        double r89852 = 2.0;
        double r89853 = r89852 * r89845;
        double r89854 = r89851 / r89853;
        return r89854;
}

↓

double f(double a, double b, double c) {
        double r89855 = b;
        double r89856 = -2.3202538172935113e+68;
        bool r89857 = r89855 <= r89856;
        double r89858 = 1.0;
        double r89859 = 2.0;
        double r89860 = r89858 / r89859;
        double r89861 = c;
        double r89862 = r89861 / r89855;
        double r89863 = a;
        double r89864 = r89855 / r89863;
        double r89865 = r89862 - r89864;
        double r89866 = r89859 * r89865;
        double r89867 = r89860 * r89866;
        double r89868 = -1.5018009284516093e-154;
        bool r89869 = r89855 <= r89868;
        double r89870 = -r89855;
        double r89871 = r89855 * r89855;
        double r89872 = 4.0;
        double r89873 = r89872 * r89863;
        double r89874 = r89873 * r89861;
        double r89875 = r89871 - r89874;
        double r89876 = sqrt(r89875);
        double r89877 = r89870 + r89876;
        double r89878 = r89859 * r89863;
        double r89879 = r89858 / r89878;
        double r89880 = r89877 * r89879;
        double r89881 = 5.026817155457435e+125;
        bool r89882 = r89855 <= r89881;
        double r89883 = r89870 - r89876;
        double r89884 = r89861 / r89883;
        double r89885 = r89884 * r89872;
        double r89886 = r89860 * r89885;
        double r89887 = -2.0;
        double r89888 = r89887 * r89862;
        double r89889 = r89860 * r89888;
        double r89890 = r89882 ? r89886 : r89889;
        double r89891 = r89869 ? r89880 : r89890;
        double r89892 = r89857 ? r89867 : r89891;
        return r89892;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.7
Target	20.9
Herbie	6.9

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

Derivation

1. Split input into 4 regimes

2. if b < -2.3202538172935113e+68

   1. Initial program 40.7

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

   3. Applied flip-+ 61.9

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

   4. Simplified 61.1

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

   6. Applied *-un-lft-identity 61.1

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

   7. Applied *-un-lft-identity 61.1

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

   8. Applied times-frac 61.1

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

   9. Applied times-frac 61.1

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

   10. Simplified 61.1

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

   11. Simplified 61.9

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

   13. Applied clear-num 61.9

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

   14. Simplified 60.9

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

   15. Taylor expanded around -inf 5.2

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

   16. Simplified 5.2

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

   if -2.3202538172935113e+68 < b < -1.5018009284516093e-154

   1. Initial program 6.4

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

   3. Applied div-inv 6.5

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

   if -1.5018009284516093e-154 < b < 5.026817155457435e+125

   1. Initial program 29.6

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

   3. Applied flip-+ 29.9

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

   4. Simplified 16.7

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

   6. Applied *-un-lft-identity 16.7

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

   7. Applied *-un-lft-identity 16.7

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

   8. Applied times-frac 16.7

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

   9. Applied times-frac 16.7

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

   10. Simplified 16.7

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

   11. Simplified 21.4

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

   13. Applied clear-num 21.6

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

   14. Simplified 10.7

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

   16. Applied div-inv 10.7

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

   17. Applied add-cube-cbrt 10.7

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

   18. Applied times-frac 10.7

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

   19. Simplified 10.4

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

   20. Simplified 10.4

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

   if 5.026817155457435e+125 < b

   1. Initial program 61.6

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

   3. Applied flip-+ 61.6

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

   4. Simplified 32.7

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

   6. Applied *-un-lft-identity 32.7

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

   7. Applied *-un-lft-identity 32.7

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

   8. Applied times-frac 32.7

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

   9. Applied times-frac 32.7

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

   10. Simplified 32.7

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

   11. Simplified 32.6

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

   13. Applied clear-num 32.7

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

   14. Simplified 31.4

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

   15. Taylor expanded around inf 1.7

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.3202538172935113 \cdot 10^{68}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)\\ \mathbf{elif}\;b \le -1.5018009284516093 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 5.0268171554574348 \cdot 10^{125}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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)))