quad2p (problem 3.2.1, positive)

Average Error: 34.5 → 6.9

Time: 10.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.514236331386388029696934038334880301838 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.213463787515188210038303888822268821527 \cdot 10^{-217}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 4.945382427151992145926851206958433093126 \cdot 10^{116}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r17364 = b_2;
        double r17365 = -r17364;
        double r17366 = r17364 * r17364;
        double r17367 = a;
        double r17368 = c;
        double r17369 = r17367 * r17368;
        double r17370 = r17366 - r17369;
        double r17371 = sqrt(r17370);
        double r17372 = r17365 + r17371;
        double r17373 = r17372 / r17367;
        return r17373;
}

↓

double f(double a, double b_2, double c) {
        double r17374 = b_2;
        double r17375 = -8.514236331386388e+118;
        bool r17376 = r17374 <= r17375;
        double r17377 = 0.5;
        double r17378 = c;
        double r17379 = r17378 / r17374;
        double r17380 = r17377 * r17379;
        double r17381 = 2.0;
        double r17382 = a;
        double r17383 = r17374 / r17382;
        double r17384 = r17381 * r17383;
        double r17385 = r17380 - r17384;
        double r17386 = -6.213463787515188e-217;
        bool r17387 = r17374 <= r17386;
        double r17388 = 1.0;
        double r17389 = r17374 * r17374;
        double r17390 = r17382 * r17378;
        double r17391 = r17389 - r17390;
        double r17392 = sqrt(r17391);
        double r17393 = r17392 - r17374;
        double r17394 = r17382 / r17393;
        double r17395 = r17388 / r17394;
        double r17396 = 4.945382427151992e+116;
        bool r17397 = r17374 <= r17396;
        double r17398 = -r17374;
        double r17399 = r17398 - r17392;
        double r17400 = r17378 / r17399;
        double r17401 = -0.5;
        double r17402 = r17401 * r17379;
        double r17403 = r17397 ? r17400 : r17402;
        double r17404 = r17387 ? r17395 : r17403;
        double r17405 = r17376 ? r17385 : r17404;
        return r17405;
}

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 < -8.514236331386388e+118

   1. Initial program 51.5

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

   2. Taylor expanded around -inf 2.8

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

   if -8.514236331386388e+118 < b_2 < -6.213463787515188e-217

   1. Initial program 8.1

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

   3. Applied clear-num 8.2

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

   4. Simplified 8.2

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

   if -6.213463787515188e-217 < b_2 < 4.945382427151992e+116

   1. Initial program 31.4

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

   3. Applied flip-+ 31.5

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

   4. Simplified 17.4

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

   6. Applied *-un-lft-identity 17.4

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

   7. Applied *-un-lft-identity 17.4

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

   8. Applied times-frac 17.4

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

   9. Simplified 17.4

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

   10. Simplified 10.1

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

   if 4.945382427151992e+116 < b_2

   1. Initial program 60.2

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

   2. Taylor expanded around inf 2.5

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.514236331386388029696934038334880301838 \cdot 10^{118}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le -6.213463787515188210038303888822268821527 \cdot 10^{-217}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 4.945382427151992145926851206958433093126 \cdot 10^{116}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))