Average Error: 1.7 → 1.7
Time: 33.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{4}{y} + \frac{x}{y}\right) - z \cdot \frac{x}{y}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{4}{y} + \frac{x}{y}\right) - z \cdot \frac{x}{y}\right|
double f(double x, double y, double z) {
        double r699165 = x;
        double r699166 = 4.0;
        double r699167 = r699165 + r699166;
        double r699168 = y;
        double r699169 = r699167 / r699168;
        double r699170 = r699165 / r699168;
        double r699171 = z;
        double r699172 = r699170 * r699171;
        double r699173 = r699169 - r699172;
        double r699174 = fabs(r699173);
        return r699174;
}
↓
double f(double x, double y, double z) {
        double r699175 = 4.0;
        double r699176 = y;
        double r699177 = r699175 / r699176;
        double r699178 = x;
        double r699179 = r699178 / r699176;
        double r699180 = r699177 + r699179;
        double r699181 = z;
        double r699182 = r699181 * r699179;
        double r699183 = r699180 - r699182;
        double r699184 = fabs(r699183);
        return r699184;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Final simplification1.7

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

Reproduce

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