NMSE problem 3.2.1

Average Error: 34.2 → 10.4

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r103479 = b_2;
        double r103480 = -r103479;
        double r103481 = r103479 * r103479;
        double r103482 = a;
        double r103483 = c;
        double r103484 = r103482 * r103483;
        double r103485 = r103481 - r103484;
        double r103486 = sqrt(r103485);
        double r103487 = r103480 - r103486;
        double r103488 = r103487 / r103482;
        return r103488;
}

↓

double f(double a, double b_2, double c) {
        double r103489 = b_2;
        double r103490 = -4.4270058556435274e-117;
        bool r103491 = r103489 <= r103490;
        double r103492 = -0.5;
        double r103493 = c;
        double r103494 = r103493 / r103489;
        double r103495 = r103492 * r103494;
        double r103496 = 2.4992282662840617e+84;
        bool r103497 = r103489 <= r103496;
        double r103498 = -r103489;
        double r103499 = r103489 * r103489;
        double r103500 = a;
        double r103501 = r103500 * r103493;
        double r103502 = r103499 - r103501;
        double r103503 = sqrt(r103502);
        double r103504 = r103498 - r103503;
        double r103505 = r103504 / r103500;
        double r103506 = 0.5;
        double r103507 = r103506 * r103494;
        double r103508 = 2.0;
        double r103509 = r103489 / r103500;
        double r103510 = r103508 * r103509;
        double r103511 = r103507 - r103510;
        double r103512 = r103497 ? r103505 : r103511;
        double r103513 = r103491 ? r103495 : r103512;
        return r103513;
}

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 < -4.4270058556435274e-117

   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 11.0

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

   if -4.4270058556435274e-117 < b_2 < 2.4992282662840617e+84

   1. Initial program 12.4

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

   3. Applied div-inv 12.5

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

   5. Applied un-div-inv 12.4

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

   if 2.4992282662840617e+84 < b_2

   1. Initial program 43.1

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

   2. Taylor expanded around inf 4.1

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.4270058556435274 \cdot 10^{-117}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.49922826628406174 \cdot 10^{84}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))