quad2p (problem 3.2.1, positive)

Average Error: 34.2 → 6.6

Time: 5.9s

Precision: 64

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.5254699676315931 \cdot 10^{122}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.27850456875614525 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.23248892134249817 \cdot 10^{79}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r20830 = b_2;
        double r20831 = -r20830;
        double r20832 = r20830 * r20830;
        double r20833 = a;
        double r20834 = c;
        double r20835 = r20833 * r20834;
        double r20836 = r20832 - r20835;
        double r20837 = sqrt(r20836);
        double r20838 = r20831 + r20837;
        double r20839 = r20838 / r20833;
        return r20839;
}

↓

double f(double a, double b_2, double c) {
        double r20840 = b_2;
        double r20841 = -1.5254699676315931e+122;
        bool r20842 = r20840 <= r20841;
        double r20843 = 0.5;
        double r20844 = c;
        double r20845 = r20844 / r20840;
        double r20846 = r20843 * r20845;
        double r20847 = 2.0;
        double r20848 = a;
        double r20849 = r20840 / r20848;
        double r20850 = r20847 * r20849;
        double r20851 = r20846 - r20850;
        double r20852 = -6.278504568756145e-182;
        bool r20853 = r20840 <= r20852;
        double r20854 = -r20840;
        double r20855 = r20840 * r20840;
        double r20856 = r20848 * r20844;
        double r20857 = r20855 - r20856;
        double r20858 = sqrt(r20857);
        double r20859 = r20854 + r20858;
        double r20860 = 1.0;
        double r20861 = r20860 / r20848;
        double r20862 = r20859 * r20861;
        double r20863 = 5.232488921342498e+79;
        bool r20864 = r20840 <= r20863;
        double r20865 = r20854 - r20858;
        double r20866 = r20860 * r20865;
        double r20867 = r20866 / r20844;
        double r20868 = r20860 / r20867;
        double r20869 = -0.5;
        double r20870 = r20869 * r20845;
        double r20871 = r20864 ? r20868 : r20870;
        double r20872 = r20853 ? r20862 : r20871;
        double r20873 = r20842 ? r20851 : r20872;
        return r20873;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.5254699676315931e+122

   1. Initial program 52.5

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 2.7

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   if -1.5254699676315931e+122 < b_2 < -6.278504568756145e-182

   1. Initial program 7.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied div-inv 7.2

      \[\leadsto \color{blue}{\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

   if -6.278504568756145e-182 < b_2 < 5.232488921342498e+79

   1. Initial program 27.8

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied flip-+ 27.9

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   4. Simplified 16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.1

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{a}\]

   7. Applied *-un-lft-identity 16.1

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + a \cdot c\right)}}{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{a}\]

   8. Applied times-frac 16.1

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

   9. Simplified 16.1

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{0 + a \cdot c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   10. Simplified 14.1

       \[\leadsto \frac{1 \cdot \color{blue}{\frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}{a}\]
   11. Using strategy rm

   12. Applied clear-num 14.1

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{1 \cdot \frac{a}{\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{c}}}}}\]

   13. Simplified 10.4

       \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}}\]

   if 5.232488921342498e+79 < b_2

   1. Initial program 58.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 2.9

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.5254699676315931 \cdot 10^{122}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.27850456875614525 \cdot 10^{-182}:\\ \;\;\;\;\left(\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \le 5.23248892134249817 \cdot 10^{79}:\\ \;\;\;\;\frac{1}{\frac{1 \cdot \left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))