Average Error: 1.7 → 0.2
Time: 3.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.141065750945845758760211222436365665667 \cdot 10^{-15} \lor \neg \left(x \le 4.40041470183000606091280929473849033149 \cdot 10^{-62}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.141065750945845758760211222436365665667 \cdot 10^{-15} \lor \neg \left(x \le 4.40041470183000606091280929473849033149 \cdot 10^{-62}\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 r31001 = x;
        double r31002 = 4.0;
        double r31003 = r31001 + r31002;
        double r31004 = y;
        double r31005 = r31003 / r31004;
        double r31006 = r31001 / r31004;
        double r31007 = z;
        double r31008 = r31006 * r31007;
        double r31009 = r31005 - r31008;
        double r31010 = fabs(r31009);
        return r31010;
}
↓
double f(double x, double y, double z) {
        double r31011 = x;
        double r31012 = -3.1410657509458458e-15;
        bool r31013 = r31011 <= r31012;
        double r31014 = 4.400414701830006e-62;
        bool r31015 = r31011 <= r31014;
        double r31016 = !r31015;
        bool r31017 = r31013 || r31016;
        double r31018 = 4.0;
        double r31019 = r31011 + r31018;
        double r31020 = y;
        double r31021 = r31019 / r31020;
        double r31022 = z;
        double r31023 = r31022 / r31020;
        double r31024 = r31011 * r31023;
        double r31025 = r31021 - r31024;
        double r31026 = fabs(r31025);
        double r31027 = r31011 * r31022;
        double r31028 = r31019 - r31027;
        double r31029 = r31028 / r31020;
        double r31030 = fabs(r31029);
        double r31031 = r31017 ? r31026 : r31030;
        return r31031;
}

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 < -3.1410657509458458e-15 or 4.400414701830006e-62 < x

    1. Initial program 0.2

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

    3. Applied div-inv0.3

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

    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 -3.1410657509458458e-15 < x < 4.400414701830006e-62

    1. Initial program 2.9

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.141065750945845758760211222436365665667 \cdot 10^{-15} \lor \neg \left(x \le 4.40041470183000606091280929473849033149 \cdot 10^{-62}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))