quad2m (problem 3.2.1, negative)

Average Error: 32.8 → 10.2

Time: 15.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.031575300615258 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.9089378078751267 \cdot 10^{+122}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r367472 = b_2;
        double r367473 = -r367472;
        double r367474 = r367472 * r367472;
        double r367475 = a;
        double r367476 = c;
        double r367477 = r367475 * r367476;
        double r367478 = r367474 - r367477;
        double r367479 = sqrt(r367478);
        double r367480 = r367473 - r367479;
        double r367481 = r367480 / r367475;
        return r367481;
}

↓

double f(double a, double b_2, double c) {
        double r367482 = b_2;
        double r367483 = -3.031575300615258e-39;
        bool r367484 = r367482 <= r367483;
        double r367485 = -0.5;
        double r367486 = c;
        double r367487 = r367486 / r367482;
        double r367488 = r367485 * r367487;
        double r367489 = 1.9089378078751267e+122;
        bool r367490 = r367482 <= r367489;
        double r367491 = -r367482;
        double r367492 = a;
        double r367493 = r367491 / r367492;
        double r367494 = r367482 * r367482;
        double r367495 = r367486 * r367492;
        double r367496 = r367494 - r367495;
        double r367497 = sqrt(r367496);
        double r367498 = r367497 / r367492;
        double r367499 = r367493 - r367498;
        double r367500 = 0.5;
        double r367501 = r367487 * r367500;
        double r367502 = 2.0;
        double r367503 = r367482 / r367492;
        double r367504 = r367502 * r367503;
        double r367505 = r367501 - r367504;
        double r367506 = r367490 ? r367499 : r367505;
        double r367507 = r367484 ? r367488 : r367506;
        return r367507;
}

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 < -3.031575300615258e-39

   1. Initial program 53.3

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

   3. Applied div-sub 53.9

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

   4. Taylor expanded around -inf 7.8

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

   if -3.031575300615258e-39 < b_2 < 1.9089378078751267e+122

   1. Initial program 13.7

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

   3. Applied div-sub 13.7

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

   if 1.9089378078751267e+122 < b_2

   1. Initial program 50.0

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

   2. Taylor expanded around inf 3.4

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.031575300615258 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.9089378078751267 \cdot 10^{+122}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))