quad2p (problem 3.2.1, positive)

Average Error: 33.8 → 10.4

Time: 14.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r30795 = b_2;
        double r30796 = -r30795;
        double r30797 = r30795 * r30795;
        double r30798 = a;
        double r30799 = c;
        double r30800 = r30798 * r30799;
        double r30801 = r30797 - r30800;
        double r30802 = sqrt(r30801);
        double r30803 = r30796 + r30802;
        double r30804 = r30803 / r30798;
        return r30804;
}

↓

double f(double a, double b_2, double c) {
        double r30805 = b_2;
        double r30806 = -3.4127765686872833e+126;
        bool r30807 = r30805 <= r30806;
        double r30808 = c;
        double r30809 = r30808 / r30805;
        double r30810 = 0.5;
        double r30811 = a;
        double r30812 = r30805 / r30811;
        double r30813 = -2.0;
        double r30814 = r30812 * r30813;
        double r30815 = fma(r30809, r30810, r30814);
        double r30816 = 4.603517726908401e-74;
        bool r30817 = r30805 <= r30816;
        double r30818 = 1.0;
        double r30819 = r30805 * r30805;
        double r30820 = r30811 * r30808;
        double r30821 = r30819 - r30820;
        double r30822 = sqrt(r30821);
        double r30823 = r30822 - r30805;
        double r30824 = r30811 / r30823;
        double r30825 = r30818 / r30824;
        double r30826 = -0.5;
        double r30827 = r30826 * r30809;
        double r30828 = r30817 ? r30825 : r30827;
        double r30829 = r30807 ? r30815 : r30828;
        return r30829;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.4127765686872833e+126

   1. Initial program 53.6

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

   2. Taylor expanded around -inf 3.2

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

   3. Simplified 3.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)}\]

   if -3.4127765686872833e+126 < b_2 < 4.603517726908401e-74

   1. Initial program 13.2

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

   3. Applied clear-num 13.3

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

   4. Simplified 13.3

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

   if 4.603517726908401e-74 < b_2

   1. Initial program 53.1

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

   2. Taylor expanded around inf 9.1

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.412776568687283300932456834981587297891 \cdot 10^{126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 4.603517726908400645968266248286182254745 \cdot 10^{-74}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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))