The quadratic formula (r1)

Average Error: 34.6 → 7.5

Time: 55.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -4.718078597954240507360630293401646945929 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.132821374632338820562375169502615703862 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r193797 = b;
        double r193798 = -r193797;
        double r193799 = r193797 * r193797;
        double r193800 = 4.0;
        double r193801 = a;
        double r193802 = r193800 * r193801;
        double r193803 = c;
        double r193804 = r193802 * r193803;
        double r193805 = r193799 - r193804;
        double r193806 = sqrt(r193805);
        double r193807 = r193798 + r193806;
        double r193808 = 2.0;
        double r193809 = r193808 * r193801;
        double r193810 = r193807 / r193809;
        return r193810;
}

↓

double f(double a, double b, double c) {
        double r193811 = b;
        double r193812 = -7.943482039519134e+75;
        bool r193813 = r193811 <= r193812;
        double r193814 = c;
        double r193815 = r193814 / r193811;
        double r193816 = a;
        double r193817 = r193811 / r193816;
        double r193818 = r193815 - r193817;
        double r193819 = 1.0;
        double r193820 = r193818 * r193819;
        double r193821 = -4.7180785979542405e-288;
        bool r193822 = r193811 <= r193821;
        double r193823 = r193811 * r193811;
        double r193824 = 4.0;
        double r193825 = r193824 * r193816;
        double r193826 = r193825 * r193814;
        double r193827 = r193823 - r193826;
        double r193828 = sqrt(r193827);
        double r193829 = -r193811;
        double r193830 = r193828 + r193829;
        double r193831 = 2.0;
        double r193832 = r193816 * r193831;
        double r193833 = r193830 / r193832;
        double r193834 = 1.1328213746323388e+81;
        bool r193835 = r193811 <= r193834;
        double r193836 = r193814 * r193816;
        double r193837 = r193824 * r193836;
        double r193838 = r193823 - r193837;
        double r193839 = sqrt(r193838);
        double r193840 = r193829 - r193839;
        double r193841 = cbrt(r193840);
        double r193842 = r193841 / r193814;
        double r193843 = cbrt(r193842);
        double r193844 = r193816 / r193843;
        double r193845 = r193844 / r193841;
        double r193846 = r193829 - r193828;
        double r193847 = cbrt(r193846);
        double r193848 = r193845 / r193847;
        double r193849 = r193848 / r193816;
        double r193850 = r193843 * r193843;
        double r193851 = r193824 / r193850;
        double r193852 = r193851 / r193831;
        double r193853 = r193849 * r193852;
        double r193854 = -1.0;
        double r193855 = r193814 * r193854;
        double r193856 = r193855 / r193811;
        double r193857 = r193835 ? r193853 : r193856;
        double r193858 = r193822 ? r193833 : r193857;
        double r193859 = r193813 ? r193820 : r193858;
        return r193859;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.0
Herbie	7.5

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

Derivation

1. Split input into 4 regimes

2. if b < -7.943482039519134e+75

   1. Initial program 42.7

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

   2. Taylor expanded around -inf 4.2

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

   3. Simplified 4.2

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

   if -7.943482039519134e+75 < b < -4.7180785979542405e-288

   1. Initial program 9.3

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

   if -4.7180785979542405e-288 < b < 1.1328213746323388e+81

   1. Initial program 30.8

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

   3. Applied flip-+ 30.8

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

   4. Simplified 16.0

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

   6. Applied add-cube-cbrt 16.7

      \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\color{blue}{\left(\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

   7. Applied associate-/r* 16.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

   8. Simplified 16.0

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

   10. Applied *-un-lft-identity 16.0

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

   11. Applied cbrt-prod 16.0

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

   12. Applied *-un-lft-identity 16.0

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

   13. Applied cbrt-prod 16.0

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

   14. Applied add-cube-cbrt 16.2

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

   15. Applied times-frac 16.2

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

   16. Applied times-frac 15.6

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

   17. Applied times-frac 15.2

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

   18. Applied times-frac 12.0

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

   if 1.1328213746323388e+81 < b

   1. Initial program 59.0

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

   2. Taylor expanded around inf 2.5

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

   3. Simplified 2.5

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

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.943482039519133630405882994043698433958 \cdot 10^{75}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le -4.718078597954240507360630293401646945929 \cdot 10^{-288}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.132821374632338820562375169502615703862 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}}{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a} \cdot \frac{\frac{4}{\sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}}{c}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))