Average Error: 1.6 → 1.7
Time: 11.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{x}{y} - \frac{z}{\frac{y}{x}}\right) + \frac{4}{y}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{x}{y} - \frac{z}{\frac{y}{x}}\right) + \frac{4}{y}\right|
double f(double x, double y, double z) {
        double r1984693 = x;
        double r1984694 = 4.0;
        double r1984695 = r1984693 + r1984694;
        double r1984696 = y;
        double r1984697 = r1984695 / r1984696;
        double r1984698 = r1984693 / r1984696;
        double r1984699 = z;
        double r1984700 = r1984698 * r1984699;
        double r1984701 = r1984697 - r1984700;
        double r1984702 = fabs(r1984701);
        return r1984702;
}
↓
double f(double x, double y, double z) {
        double r1984703 = x;
        double r1984704 = y;
        double r1984705 = r1984703 / r1984704;
        double r1984706 = z;
        double r1984707 = r1984704 / r1984703;
        double r1984708 = r1984706 / r1984707;
        double r1984709 = r1984705 - r1984708;
        double r1984710 = 4.0;
        double r1984711 = r1984710 / r1984704;
        double r1984712 = r1984709 + r1984711;
        double r1984713 = fabs(r1984712);
        return r1984713;
}

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 1.6

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

  2. Taylor expanded around 0 3.0

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

  3. Simplified1.6

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

  4. Taylor expanded around 0 3.0

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

  5. Simplified1.7

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

  6. Final simplification1.7

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

Reproduce

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