NMSE problem 3.2.1

Average Error: 34.5 → 10.1

Time: 4.0s

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 r103621 = b_2;
        double r103622 = -r103621;
        double r103623 = r103621 * r103621;
        double r103624 = a;
        double r103625 = c;
        double r103626 = r103624 * r103625;
        double r103627 = r103623 - r103626;
        double r103628 = sqrt(r103627);
        double r103629 = r103622 - r103628;
        double r103630 = r103629 / r103624;
        return r103630;
}

↓

double f(double a, double b_2, double c) {
        double r103631 = b_2;
        double r103632 = -4.706781135059312e-92;
        bool r103633 = r103631 <= r103632;
        double r103634 = -0.5;
        double r103635 = c;
        double r103636 = r103635 / r103631;
        double r103637 = r103634 * r103636;
        double r103638 = 5.722235152988638e+98;
        bool r103639 = r103631 <= r103638;
        double r103640 = -r103631;
        double r103641 = r103631 * r103631;
        double r103642 = a;
        double r103643 = r103642 * r103635;
        double r103644 = r103641 - r103643;
        double r103645 = sqrt(r103644);
        double r103646 = r103640 - r103645;
        double r103647 = r103646 / r103642;
        double r103648 = 0.5;
        double r103649 = r103648 * r103636;
        double r103650 = 2.0;
        double r103651 = r103631 / r103642;
        double r103652 = r103650 * r103651;
        double r103653 = r103649 - r103652;
        double r103654 = r103639 ? r103647 : r103653;
        double r103655 = r103633 ? r103637 : r103654;
        return r103655;
}

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 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))