NMSE problem 3.2.1

Average Error: 34.1 → 10.5

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r73708 = b_2;
        double r73709 = -r73708;
        double r73710 = r73708 * r73708;
        double r73711 = a;
        double r73712 = c;
        double r73713 = r73711 * r73712;
        double r73714 = r73710 - r73713;
        double r73715 = sqrt(r73714);
        double r73716 = r73709 - r73715;
        double r73717 = r73716 / r73711;
        return r73717;
}

↓

double f(double a, double b_2, double c) {
        double r73718 = b_2;
        double r73719 = -5.6439472304372656e-71;
        bool r73720 = r73718 <= r73719;
        double r73721 = -0.5;
        double r73722 = c;
        double r73723 = r73722 / r73718;
        double r73724 = r73721 * r73723;
        double r73725 = 1.4983382189642058e+54;
        bool r73726 = r73718 <= r73725;
        double r73727 = 1.0;
        double r73728 = a;
        double r73729 = -r73718;
        double r73730 = r73718 * r73718;
        double r73731 = r73728 * r73722;
        double r73732 = r73730 - r73731;
        double r73733 = sqrt(r73732);
        double r73734 = r73729 - r73733;
        double r73735 = r73728 / r73734;
        double r73736 = r73727 / r73735;
        double r73737 = 0.5;
        double r73738 = r73737 * r73723;
        double r73739 = 2.0;
        double r73740 = r73718 / r73728;
        double r73741 = r73739 * r73740;
        double r73742 = r73738 - r73741;
        double r73743 = r73726 ? r73736 : r73742;
        double r73744 = r73720 ? r73724 : r73743;
        return r73744;
}

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 < -5.6439472304372656e-71

   1. Initial program 53.3

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

   2. Taylor expanded around -inf 9.1

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

   if -5.6439472304372656e-71 < b_2 < 1.4983382189642058e+54

   1. Initial program 14.2

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

   3. Applied clear-num 14.3

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

   if 1.4983382189642058e+54 < b_2

   1. Initial program 38.0

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

   2. Taylor expanded around inf 5.0

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.643947230437265585428917170074785083411 \cdot 10^{-71}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.498338218964205825262884582884276173268 \cdot 10^{54}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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))