Average Error: 1.6 → 0.5
Time: 3.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -71921344804687528:\\ \;\;\;\;\left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 9.36037578809800899 \cdot 10^{-141}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -71921344804687528:\\
\;\;\;\;\left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 9.36037578809800899 \cdot 10^{-141}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\end{array}
double f(double x, double y, double z) {
        double r29113 = x;
        double r29114 = 4.0;
        double r29115 = r29113 + r29114;
        double r29116 = y;
        double r29117 = r29115 / r29116;
        double r29118 = r29113 / r29116;
        double r29119 = z;
        double r29120 = r29118 * r29119;
        double r29121 = r29117 - r29120;
        double r29122 = fabs(r29121);
        return r29122;
}
↓
double f(double x, double y, double z) {
        double r29123 = x;
        double r29124 = -7.192134480468753e+16;
        bool r29125 = r29123 <= r29124;
        double r29126 = 4.0;
        double r29127 = 1.0;
        double r29128 = y;
        double r29129 = r29127 / r29128;
        double r29130 = r29126 * r29129;
        double r29131 = r29123 / r29128;
        double r29132 = r29130 + r29131;
        double r29133 = z;
        double r29134 = r29133 / r29128;
        double r29135 = r29123 * r29134;
        double r29136 = r29132 - r29135;
        double r29137 = fabs(r29136);
        double r29138 = 9.360375788098009e-141;
        bool r29139 = r29123 <= r29138;
        double r29140 = r29123 + r29126;
        double r29141 = r29123 * r29133;
        double r29142 = r29140 - r29141;
        double r29143 = r29142 / r29128;
        double r29144 = fabs(r29143);
        double r29145 = r29131 * r29133;
        double r29146 = r29132 - r29145;
        double r29147 = fabs(r29146);
        double r29148 = r29139 ? r29144 : r29147;
        double r29149 = r29125 ? r29137 : r29148;
        return r29149;
}

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 < -7.192134480468753e+16

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    4. Applied div-inv0.2

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

    5. Applied associate-*l*0.2

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

    6. Simplified0.1

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

    if -7.192134480468753e+16 < x < 9.360375788098009e-141

    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

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

    if 9.360375788098009e-141 < x

    1. Initial program 1.3

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \left|\color{blue}{\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right)} - \frac{x}{y} \cdot z\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -71921344804687528:\\ \;\;\;\;\left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 9.36037578809800899 \cdot 10^{-141}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(4 \cdot \frac{1}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))