Average Error: 1.6 → 0.2
Time: 16.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.225256278246885 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 1.1828035466780113 \cdot 10^{+84}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.225256278246885 \cdot 10^{+40}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 1.1828035466780113 \cdot 10^{+84}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r870158 = x;
        double r870159 = 4.0;
        double r870160 = r870158 + r870159;
        double r870161 = y;
        double r870162 = r870160 / r870161;
        double r870163 = r870158 / r870161;
        double r870164 = z;
        double r870165 = r870163 * r870164;
        double r870166 = r870162 - r870165;
        double r870167 = fabs(r870166);
        return r870167;
}
↓
double f(double x, double y, double z) {
        double r870168 = x;
        double r870169 = -2.225256278246885e+40;
        bool r870170 = r870168 <= r870169;
        double r870171 = 4.0;
        double r870172 = r870171 + r870168;
        double r870173 = y;
        double r870174 = r870172 / r870173;
        double r870175 = z;
        double r870176 = r870175 / r870173;
        double r870177 = r870168 * r870176;
        double r870178 = r870174 - r870177;
        double r870179 = fabs(r870178);
        double r870180 = 1.1828035466780113e+84;
        bool r870181 = r870168 <= r870180;
        double r870182 = r870175 * r870168;
        double r870183 = r870172 - r870182;
        double r870184 = r870183 / r870173;
        double r870185 = fabs(r870184);
        double r870186 = r870171 / r870173;
        double r870187 = r870168 / r870173;
        double r870188 = r870186 + r870187;
        double r870189 = r870173 / r870168;
        double r870190 = r870175 / r870189;
        double r870191 = r870188 - r870190;
        double r870192 = fabs(r870191);
        double r870193 = r870181 ? r870185 : r870192;
        double r870194 = r870170 ? r870179 : r870193;
        return r870194;
}

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 3 regimes

  2. if x < -2.225256278246885e+40

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

    4. Applied associate-*l*0.2

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

    5. Simplified0.1

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

    if -2.225256278246885e+40 < x < 1.1828035466780113e+84

    1. Initial program 2.2

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

    3. Applied div-inv2.3

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

    4. Applied associate-*l*4.5

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

    5. Simplified4.5

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

    7. Applied associate-*r/0.3

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

    8. Applied sub-div0.3

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

    if 1.1828035466780113e+84 < x

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

    4. Applied associate-*l*0.2

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

    5. Simplified0.1

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

    6. Taylor expanded around inf 13.2

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

    7. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.225256278246885 \cdot 10^{+40}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 1.1828035466780113 \cdot 10^{+84}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))