NMSE problem 3.2.1

Average Error: 34.5 → 10.1

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r76508 = b_2;
        double r76509 = -r76508;
        double r76510 = r76508 * r76508;
        double r76511 = a;
        double r76512 = c;
        double r76513 = r76511 * r76512;
        double r76514 = r76510 - r76513;
        double r76515 = sqrt(r76514);
        double r76516 = r76509 - r76515;
        double r76517 = r76516 / r76511;
        return r76517;
}

↓

double f(double a, double b_2, double c) {
        double r76518 = b_2;
        double r76519 = -4.706781135059312e-92;
        bool r76520 = r76518 <= r76519;
        double r76521 = -0.5;
        double r76522 = c;
        double r76523 = r76522 / r76518;
        double r76524 = r76521 * r76523;
        double r76525 = 5.722235152988638e+98;
        bool r76526 = r76518 <= r76525;
        double r76527 = -r76518;
        double r76528 = r76518 * r76518;
        double r76529 = a;
        double r76530 = r76529 * r76522;
        double r76531 = r76528 - r76530;
        double r76532 = sqrt(r76531);
        double r76533 = r76527 - r76532;
        double r76534 = r76533 / r76529;
        double r76535 = 0.5;
        double r76536 = r76535 * r76523;
        double r76537 = 2.0;
        double r76538 = r76518 / r76529;
        double r76539 = r76537 * r76538;
        double r76540 = r76536 - r76539;
        double r76541 = r76526 ? r76534 : r76540;
        double r76542 = r76520 ? r76524 : r76541;
        return r76542;
}

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 < -4.706781135059312e-92

   1. Initial program 52.4

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

   2. Taylor expanded around -inf 10.2

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

   if -4.706781135059312e-92 < b_2 < 5.722235152988638e+98

   1. Initial program 12.6

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

   3. Applied div-inv 12.7

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

   5. Applied un-div-inv 12.6

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

   if 5.722235152988638e+98 < b_2

   1. Initial program 47.1

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

   2. Taylor expanded around inf 3.6

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.706781135059311758856471716413486308072 \cdot 10^{-92}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 5.722235152988638272816037483919181313619 \cdot 10^{98}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))