Average Error: 1.5 → 0.4
Time: 13.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -16246640829.1726360321044921875:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\end{array}
double f(double x, double y, double z) {
        double r1170123 = x;
        double r1170124 = 4.0;
        double r1170125 = r1170123 + r1170124;
        double r1170126 = y;
        double r1170127 = r1170125 / r1170126;
        double r1170128 = r1170123 / r1170126;
        double r1170129 = z;
        double r1170130 = r1170128 * r1170129;
        double r1170131 = r1170127 - r1170130;
        double r1170132 = fabs(r1170131);
        return r1170132;
}
↓
double f(double x, double y, double z) {
        double r1170133 = x;
        double r1170134 = -16246640829.172636;
        bool r1170135 = r1170133 <= r1170134;
        double r1170136 = 4.0;
        double r1170137 = y;
        double r1170138 = r1170136 / r1170137;
        double r1170139 = r1170133 / r1170137;
        double r1170140 = r1170138 + r1170139;
        double r1170141 = z;
        double r1170142 = r1170139 * r1170141;
        double r1170143 = r1170140 - r1170142;
        double r1170144 = fabs(r1170143);
        double r1170145 = 1.0296033599100243e-161;
        bool r1170146 = r1170133 <= r1170145;
        double r1170147 = r1170136 + r1170133;
        double r1170148 = r1170133 * r1170141;
        double r1170149 = r1170147 - r1170148;
        double r1170150 = r1170149 / r1170137;
        double r1170151 = fabs(r1170150);
        double r1170152 = r1170146 ? r1170151 : r1170144;
        double r1170153 = r1170135 ? r1170144 : r1170152;
        return r1170153;
}

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. Split input into 2 regimes

  2. if x < -16246640829.172636 or 1.0296033599100243e-161 < x

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    if -16246640829.172636 < x < 1.0296033599100243e-161

    1. Initial program 2.5

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

      \[\leadsto \left|\color{blue}{\frac{\left(x + 4\right) - x \cdot z}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))