quad2p (problem 3.2.1, positive)

Average Error: 33.7 → 10.8

Time: 24.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.61268387266151013 \cdot 10^{141}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r77591 = b_2;
        double r77592 = -r77591;
        double r77593 = r77591 * r77591;
        double r77594 = a;
        double r77595 = c;
        double r77596 = r77594 * r77595;
        double r77597 = r77593 - r77596;
        double r77598 = sqrt(r77597);
        double r77599 = r77592 + r77598;
        double r77600 = r77599 / r77594;
        return r77600;
}

↓

double f(double a, double b_2, double c) {
        double r77601 = b_2;
        double r77602 = -2.61268387266151e+141;
        bool r77603 = r77601 <= r77602;
        double r77604 = 0.5;
        double r77605 = c;
        double r77606 = r77605 / r77601;
        double r77607 = r77604 * r77606;
        double r77608 = a;
        double r77609 = r77601 / r77608;
        double r77610 = r77607 - r77609;
        double r77611 = r77610 - r77609;
        double r77612 = 1.1860189201379418e-161;
        bool r77613 = r77601 <= r77612;
        double r77614 = r77601 * r77601;
        double r77615 = r77608 * r77605;
        double r77616 = r77614 - r77615;
        double r77617 = sqrt(r77616);
        double r77618 = r77617 / r77608;
        double r77619 = r77618 - r77609;
        double r77620 = -0.5;
        double r77621 = r77620 * r77606;
        double r77622 = r77613 ? r77619 : r77621;
        double r77623 = r77603 ? r77611 : r77622;
        return r77623;
}

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 < -2.61268387266151e+141

   1. Initial program 59.4

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

   2. Simplified 59.4

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied div-sub 59.4

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

   5. Taylor expanded around -inf 2.9

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

   if -2.61268387266151e+141 < b_2 < 1.1860189201379418e-161

   1. Initial program 10.2

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

   2. Simplified 10.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied div-sub 10.2

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

   if 1.1860189201379418e-161 < b_2

   1. Initial program 49.6

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

   2. Simplified 49.6

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

   3. Taylor expanded around inf 13.6

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

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.61268387266151013 \cdot 10^{141}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a}\right) - \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.1860189201379418 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))