The quadratic formula (r1)

Average Error: 34.4 → 6.8

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\frac{4}{1} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r81805 = b;
        double r81806 = -r81805;
        double r81807 = r81805 * r81805;
        double r81808 = 4.0;
        double r81809 = a;
        double r81810 = r81808 * r81809;
        double r81811 = c;
        double r81812 = r81810 * r81811;
        double r81813 = r81807 - r81812;
        double r81814 = sqrt(r81813);
        double r81815 = r81806 + r81814;
        double r81816 = 2.0;
        double r81817 = r81816 * r81809;
        double r81818 = r81815 / r81817;
        return r81818;
}

↓

double f(double a, double b, double c) {
        double r81819 = b;
        double r81820 = -4.301018409236461e+98;
        bool r81821 = r81819 <= r81820;
        double r81822 = 1.0;
        double r81823 = c;
        double r81824 = r81823 / r81819;
        double r81825 = a;
        double r81826 = r81819 / r81825;
        double r81827 = r81824 - r81826;
        double r81828 = r81822 * r81827;
        double r81829 = -2.3757722518657493e-260;
        bool r81830 = r81819 <= r81829;
        double r81831 = -r81819;
        double r81832 = r81819 * r81819;
        double r81833 = 4.0;
        double r81834 = r81833 * r81825;
        double r81835 = r81834 * r81823;
        double r81836 = r81832 - r81835;
        double r81837 = sqrt(r81836);
        double r81838 = r81831 + r81837;
        double r81839 = 1.0;
        double r81840 = 2.0;
        double r81841 = r81840 * r81825;
        double r81842 = r81839 / r81841;
        double r81843 = r81838 * r81842;
        double r81844 = 6.6664567809045535e+68;
        bool r81845 = r81819 <= r81844;
        double r81846 = r81839 / r81840;
        double r81847 = r81833 / r81839;
        double r81848 = r81847 * r81823;
        double r81849 = r81846 * r81848;
        double r81850 = r81831 - r81837;
        double r81851 = r81849 / r81850;
        double r81852 = -1.0;
        double r81853 = r81852 * r81824;
        double r81854 = r81845 ? r81851 : r81853;
        double r81855 = r81830 ? r81843 : r81854;
        double r81856 = r81821 ? r81828 : r81855;
        return r81856;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.4
Target	21.5
Herbie	6.8

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

Derivation

1. Split input into 4 regimes

2. if b < -4.301018409236461e+98

   1. Initial program 47.2

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

   2. Taylor expanded around -inf 3.9

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

   3. Simplified 3.9

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

   if -4.301018409236461e+98 < b < -2.3757722518657493e-260

   1. Initial program 8.5

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

   3. Applied div-inv 8.7

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

   if -2.3757722518657493e-260 < b < 6.6664567809045535e+68

   1. Initial program 29.1

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

   3. Applied flip-+ 29.1

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

   4. Simplified 16.2

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

   6. Applied clear-num 16.4

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

   7. Simplified 16.0

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

   9. Applied associate-/r* 15.8

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

   10. Simplified 9.6

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

   12. Applied *-un-lft-identity 9.6

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

   13. Applied add-sqr-sqrt 9.6

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

   14. Applied times-frac 9.6

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

   15. Applied div-inv 9.6

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

   16. Applied add-sqr-sqrt 9.6

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

   17. Applied times-frac 9.6

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

   18. Applied times-frac 9.7

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

   19. Simplified 9.7

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

   20. Simplified 9.6

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

   if 6.6664567809045535e+68 < b

   1. Initial program 58.7

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

   2. Taylor expanded around inf 3.5

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

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.30101840923646093 \cdot 10^{98}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.37577225186574925 \cdot 10^{-260}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.66645678090455348 \cdot 10^{68}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(\frac{4}{1} \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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)))