quad2p (problem 3.2.1, positive)

Average Error: 33.4 → 9.9

Time: 18.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.056361093164439 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r315425 = b_2;
        double r315426 = -r315425;
        double r315427 = r315425 * r315425;
        double r315428 = a;
        double r315429 = c;
        double r315430 = r315428 * r315429;
        double r315431 = r315427 - r315430;
        double r315432 = sqrt(r315431);
        double r315433 = r315426 + r315432;
        double r315434 = r315433 / r315428;
        return r315434;
}

↓

double f(double a, double b_2, double c) {
        double r315435 = b_2;
        double r315436 = -1.056361093164439e+152;
        bool r315437 = r315435 <= r315436;
        double r315438 = -2.0;
        double r315439 = a;
        double r315440 = r315435 / r315439;
        double r315441 = 0.5;
        double r315442 = c;
        double r315443 = r315442 / r315435;
        double r315444 = r315441 * r315443;
        double r315445 = fma(r315438, r315440, r315444);
        double r315446 = 2.326372645943808e-74;
        bool r315447 = r315435 <= r315446;
        double r315448 = r315435 * r315435;
        double r315449 = r315442 * r315439;
        double r315450 = r315448 - r315449;
        double r315451 = sqrt(r315450);
        double r315452 = r315451 - r315435;
        double r315453 = r315452 / r315439;
        double r315454 = -0.5;
        double r315455 = r315443 * r315454;
        double r315456 = r315447 ? r315453 : r315455;
        double r315457 = r315437 ? r315445 : r315456;
        return r315457;
}

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 < -1.056361093164439e+152

   1. Initial program 59.8

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

   2. Simplified 59.8

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

   3. Taylor expanded around -inf 2.6

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

   4. Simplified 2.6

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

   if -1.056361093164439e+152 < b_2 < 2.326372645943808e-74

   1. Initial program 12.3

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

   2. Simplified 12.3

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

   3. Taylor expanded around inf 12.3

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

   4. Simplified 12.3

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

   if 2.326372645943808e-74 < b_2

   1. Initial program 52.5

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

   2. Simplified 52.5

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

   3. Taylor expanded around inf 8.8

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.056361093164439 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))