Average Error: 1.6 → 1.6
Time: 6.9m
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 r18556082 = x;
        double r18556083 = 4.0;
        double r18556084 = r18556082 + r18556083;
        double r18556085 = y;
        double r18556086 = r18556084 / r18556085;
        double r18556087 = r18556082 / r18556085;
        double r18556088 = z;
        double r18556089 = r18556087 * r18556088;
        double r18556090 = r18556086 - r18556089;
        double r18556091 = fabs(r18556090);
        return r18556091;
}
↓
double f(double x, double y, double z) {
        double r18556092 = x;
        double r18556093 = y;
        double r18556094 = r18556092 / r18556093;
        double r18556095 = 4.0;
        double r18556096 = r18556095 / r18556093;
        double r18556097 = r18556094 + r18556096;
        double r18556098 = z;
        double r18556099 = r18556098 * r18556094;
        double r18556100 = r18556097 - r18556099;
        double r18556101 = fabs(r18556100);
        return r18556101;
}

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 2019112 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))