Average Error: 1.6 → 1.6
Time: 27.1s
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 r1848083 = x;
        double r1848084 = 4.0;
        double r1848085 = r1848083 + r1848084;
        double r1848086 = y;
        double r1848087 = r1848085 / r1848086;
        double r1848088 = r1848083 / r1848086;
        double r1848089 = z;
        double r1848090 = r1848088 * r1848089;
        double r1848091 = r1848087 - r1848090;
        double r1848092 = fabs(r1848091);
        return r1848092;
}
↓
double f(double x, double y, double z) {
        double r1848093 = x;
        double r1848094 = y;
        double r1848095 = r1848093 / r1848094;
        double r1848096 = 4.0;
        double r1848097 = r1848096 / r1848094;
        double r1848098 = r1848095 + r1848097;
        double r1848099 = z;
        double r1848100 = r1848099 * r1848095;
        double r1848101 = r1848098 - r1848100;
        double r1848102 = fabs(r1848101);
        return r1848102;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.6

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

  2. Taylor expanded around 0 1.6

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

  3. Simplified1.6

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

  4. Final simplification1.6

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))