jeff quadratic root 2

Average Error: 19.8 → 6.3

Time: 17.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 7.691330759470167154555132478550401992059 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}, -b\right)}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, 2, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r35745 = b;
        double r35746 = 0.0;
        bool r35747 = r35745 >= r35746;
        double r35748 = 2.0;
        double r35749 = c;
        double r35750 = r35748 * r35749;
        double r35751 = -r35745;
        double r35752 = r35745 * r35745;
        double r35753 = 4.0;
        double r35754 = a;
        double r35755 = r35753 * r35754;
        double r35756 = r35755 * r35749;
        double r35757 = r35752 - r35756;
        double r35758 = sqrt(r35757);
        double r35759 = r35751 - r35758;
        double r35760 = r35750 / r35759;
        double r35761 = r35751 + r35758;
        double r35762 = r35748 * r35754;
        double r35763 = r35761 / r35762;
        double r35764 = r35747 ? r35760 : r35763;
        return r35764;
}

↓

double f(double a, double b, double c) {
        double r35765 = b;
        double r35766 = -3.7108875578650606e+138;
        bool r35767 = r35765 <= r35766;
        double r35768 = 0.0;
        bool r35769 = r35765 >= r35768;
        double r35770 = 2.0;
        double r35771 = c;
        double r35772 = r35770 * r35771;
        double r35773 = -r35765;
        double r35774 = 4.0;
        double r35775 = r35774 * r35771;
        double r35776 = a;
        double r35777 = -r35776;
        double r35778 = r35765 * r35765;
        double r35779 = fma(r35775, r35777, r35778);
        double r35780 = sqrt(r35779);
        double r35781 = r35773 - r35780;
        double r35782 = r35772 / r35781;
        double r35783 = -2.0;
        double r35784 = r35765 * r35783;
        double r35785 = r35784 / r35770;
        double r35786 = r35785 / r35776;
        double r35787 = r35769 ? r35782 : r35786;
        double r35788 = 7.691330759470167e+83;
        bool r35789 = r35765 <= r35788;
        double r35790 = sqrt(r35780);
        double r35791 = fma(r35790, r35790, r35773);
        double r35792 = r35791 / r35770;
        double r35793 = r35792 / r35776;
        double r35794 = r35769 ? r35782 : r35793;
        double r35795 = r35765 / r35771;
        double r35796 = r35776 / r35795;
        double r35797 = fma(r35796, r35770, r35784);
        double r35798 = r35772 / r35797;
        double r35799 = r35780 - r35765;
        double r35800 = r35799 / r35770;
        double r35801 = r35800 / r35776;
        double r35802 = r35769 ? r35798 : r35801;
        double r35803 = r35789 ? r35794 : r35802;
        double r35804 = r35767 ? r35787 : r35803;
        return r35804;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -3.7108875578650606e+138

   1. Initial program 58.5

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

   2. Simplified 58.5

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 58.5

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

   5. Applied sqrt-prod 58.5

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

   6. Applied fma-neg 58.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}, -b\right)}{2}}{a}\\ \end{array}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 58.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}}, \sqrt{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}, -b\right)}{2}}{a}\\ \end{array}\]

   9. Simplified 58.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}}, \sqrt{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}, -b\right)}{2}}{a}\\ \end{array}\]

   10. Simplified 58.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}}, \sqrt{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}, -b\right)}{2}}{a}\\ \end{array}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 58.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}}, \sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}}, -b\right)}{2}}{a}\\ \end{array}\]

   13. Simplified 58.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}}, \sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}}, -b\right)}{2}}{a}\\ \end{array}\]

   14. Simplified 58.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(b, b, \left(-a\right) \cdot \left(c \cdot 4\right)\right)}}}, \sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(-c, a \cdot 4, b \cdot b\right)}}}, -b\right)}{2}}{a}\\ \end{array}\]

   15. Taylor expanded around -inf 2.2

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

   16. Simplified 2.2

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

   if -3.7108875578650606e+138 < b < 7.691330759470167e+83

   1. Initial program 8.4

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

   2. Simplified 8.4

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 8.4

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

   5. Applied sqrt-prod 8.6

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

   6. Applied fma-neg 8.5

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

   if 7.691330759470167e+83 < b

   1. Initial program 28.2

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

   2. Simplified 28.2

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}}\]

   3. Taylor expanded around inf 6.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

   4. Simplified 2.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, 2, -2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c \cdot 4, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.710887557865060611891812934492943223731 \cdot 10^{138}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot -2}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \le 7.691330759470167154555132478550401992059 \cdot 10^{83}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)}}, -b\right)}{2}}{a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, 2, b \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(4 \cdot c, -a, b \cdot b\right)} - b}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))