quad2p (problem 3.2.1, positive)

Average Error: 34.3 → 10.3

Time: 24.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 2.482189776556312363400755841338119676243 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r17080 = b_2;
        double r17081 = -r17080;
        double r17082 = r17080 * r17080;
        double r17083 = a;
        double r17084 = c;
        double r17085 = r17083 * r17084;
        double r17086 = r17082 - r17085;
        double r17087 = sqrt(r17086);
        double r17088 = r17081 + r17087;
        double r17089 = r17088 / r17083;
        return r17089;
}

↓

double f(double a, double b_2, double c) {
        double r17090 = b_2;
        double r17091 = -1.569310777886352e+111;
        bool r17092 = r17090 <= r17091;
        double r17093 = c;
        double r17094 = r17093 / r17090;
        double r17095 = 0.5;
        double r17096 = a;
        double r17097 = r17090 / r17096;
        double r17098 = -2.0;
        double r17099 = r17097 * r17098;
        double r17100 = fma(r17094, r17095, r17099);
        double r17101 = 2.4821897765563124e-128;
        bool r17102 = r17090 <= r17101;
        double r17103 = r17090 * r17090;
        double r17104 = r17096 * r17093;
        double r17105 = r17103 - r17104;
        double r17106 = sqrt(r17105);
        double r17107 = r17106 - r17090;
        double r17108 = r17107 / r17096;
        double r17109 = -0.5;
        double r17110 = r17109 * r17094;
        double r17111 = r17102 ? r17108 : r17110;
        double r17112 = r17092 ? r17100 : r17111;
        return r17112;
}

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 < -1.569310777886352e+111

   1. Initial program 50.4

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

   2. Simplified 50.4

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

   3. Taylor expanded around -inf 3.8

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

   4. Simplified 3.8

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

   if -1.569310777886352e+111 < b_2 < 2.4821897765563124e-128

   1. Initial program 11.1

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

   2. Simplified 11.1

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

   if 2.4821897765563124e-128 < b_2

   1. Initial program 51.2

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

   2. Simplified 51.2

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

   3. Taylor expanded around inf 11.6

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.569310777886352095486911207889814773134 \cdot 10^{111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b_2}, \frac{1}{2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \le 2.482189776556312363400755841338119676243 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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))