Average Error: 1.5 → 0.3
Time: 3.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.344677228311482 \cdot 10^{84} \lor \neg \left(y \le 6.0622625159831922 \cdot 10^{-76}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.344677228311482 \cdot 10^{84} \lor \neg \left(y \le 6.0622625159831922 \cdot 10^{-76}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r30084 = x;
        double r30085 = 4.0;
        double r30086 = r30084 + r30085;
        double r30087 = y;
        double r30088 = r30086 / r30087;
        double r30089 = r30084 / r30087;
        double r30090 = z;
        double r30091 = r30089 * r30090;
        double r30092 = r30088 - r30091;
        double r30093 = fabs(r30092);
        return r30093;
}
↓
double f(double x, double y, double z) {
        double r30094 = y;
        double r30095 = -7.344677228311482e+84;
        bool r30096 = r30094 <= r30095;
        double r30097 = 6.062262515983192e-76;
        bool r30098 = r30094 <= r30097;
        double r30099 = !r30098;
        bool r30100 = r30096 || r30099;
        double r30101 = x;
        double r30102 = 4.0;
        double r30103 = r30101 + r30102;
        double r30104 = r30103 / r30094;
        double r30105 = z;
        double r30106 = r30105 / r30094;
        double r30107 = r30101 * r30106;
        double r30108 = r30104 - r30107;
        double r30109 = fabs(r30108);
        double r30110 = r30101 * r30105;
        double r30111 = r30103 - r30110;
        double r30112 = r30111 / r30094;
        double r30113 = fabs(r30112);
        double r30114 = r30100 ? r30109 : r30113;
        return r30114;
}

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 y < -7.344677228311482e+84 or 6.062262515983192e-76 < y

    1. Initial program 2.5

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

    3. Applied div-inv2.5

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

    4. Applied associate-*l*0.3

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

    5. Simplified0.3

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

    if -7.344677228311482e+84 < y < 6.062262515983192e-76

    1. Initial program 0.1

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

    3. Applied associate-*l/0.4

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

    4. Applied sub-div0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.344677228311482 \cdot 10^{84} \lor \neg \left(y \le 6.0622625159831922 \cdot 10^{-76}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))