Average Error: 1.6 → 0.1
Time: 7.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17} \lor \neg \left(x \le 4.17631854478254968 \cdot 10^{-35}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17} \lor \neg \left(x \le 4.17631854478254968 \cdot 10^{-35}\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 r31936 = x;
        double r31937 = 4.0;
        double r31938 = r31936 + r31937;
        double r31939 = y;
        double r31940 = r31938 / r31939;
        double r31941 = r31936 / r31939;
        double r31942 = z;
        double r31943 = r31941 * r31942;
        double r31944 = r31940 - r31943;
        double r31945 = fabs(r31944);
        return r31945;
}
↓
double f(double x, double y, double z) {
        double r31946 = x;
        double r31947 = -1.384698452851939e-17;
        bool r31948 = r31946 <= r31947;
        double r31949 = 4.1763185447825497e-35;
        bool r31950 = r31946 <= r31949;
        double r31951 = !r31950;
        bool r31952 = r31948 || r31951;
        double r31953 = 4.0;
        double r31954 = r31946 + r31953;
        double r31955 = y;
        double r31956 = r31954 / r31955;
        double r31957 = z;
        double r31958 = r31957 / r31955;
        double r31959 = r31946 * r31958;
        double r31960 = r31956 - r31959;
        double r31961 = fabs(r31960);
        double r31962 = r31946 * r31957;
        double r31963 = r31954 - r31962;
        double r31964 = r31963 / r31955;
        double r31965 = fabs(r31964);
        double r31966 = r31952 ? r31961 : r31965;
        return r31966;
}

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 < -1.384698452851939e-17 or 4.1763185447825497e-35 < x

    1. Initial program 0.2

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

    3. Applied div-inv0.2

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

    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. Simplified0.2

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

    if -1.384698452851939e-17 < x < 4.1763185447825497e-35

    1. Initial program 2.8

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.38469845285193904 \cdot 10^{-17} \lor \neg \left(x \le 4.17631854478254968 \cdot 10^{-35}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))