fabs fraction 1

Average Error: 1.7 → 0.1

Time: 15.0s

Precision: 64

Math

\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

↓

\[\begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 2046219022.6208431720733642578125:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r14177 = x;
        double r14178 = 4.0;
        double r14179 = r14177 + r14178;
        double r14180 = y;
        double r14181 = r14179 / r14180;
        double r14182 = r14177 / r14180;
        double r14183 = z;
        double r14184 = r14182 * r14183;
        double r14185 = r14181 - r14184;
        double r14186 = fabs(r14185);
        return r14186;
}

↓

double f(double x, double y, double z) {
        double r14187 = x;
        double r14188 = 4.0;
        double r14189 = r14187 + r14188;
        double r14190 = y;
        double r14191 = r14189 / r14190;
        double r14192 = r14187 / r14190;
        double r14193 = z;
        double r14194 = r14192 * r14193;
        double r14195 = r14191 - r14194;
        double r14196 = fabs(r14195);
        double r14197 = 2046219022.6208432;
        bool r14198 = r14196 <= r14197;
        double r14199 = r14187 * r14193;
        double r14200 = r14189 - r14199;
        double r14201 = r14200 / r14190;
        double r14202 = fabs(r14201);
        double r14203 = r14198 ? r14202 : r14196;
        return r14203;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

1. Split input into 2 regimes

2. if (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) < 2046219022.6208432

   1. Initial program 4.0

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
   2. Using strategy rm

   3. Applied associate-*l/ 0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]

   4. Applied sub-div 0.1

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]

   if 2046219022.6208432 < (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))

   1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 2046219022.6208431720733642578125:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019315 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))