Average Error: 1.7 → 1.7
Time: 11.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|
double f(double x, double y, double z) {
        double r452212 = x;
        double r452213 = 4.0;
        double r452214 = r452212 + r452213;
        double r452215 = y;
        double r452216 = r452214 / r452215;
        double r452217 = r452212 / r452215;
        double r452218 = z;
        double r452219 = r452217 * r452218;
        double r452220 = r452216 - r452219;
        double r452221 = fabs(r452220);
        return r452221;
}
↓
double f(double x, double y, double z) {
        double r452222 = x;
        double r452223 = y;
        double r452224 = r452222 / r452223;
        double r452225 = 4.0;
        double r452226 = r452225 / r452223;
        double r452227 = r452224 + r452226;
        double r452228 = z;
        double r452229 = r452228 * r452224;
        double r452230 = r452227 - r452229;
        double r452231 = fabs(r452230);
        return r452231;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Derivation

  1. Initial program 1.7

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

  2. Taylor expanded around inf 1.7

    \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right)} - \frac{x}{y} \cdot z\right|\]

  3. Simplified1.7

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

  4. Final simplification1.7

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))