Average Error: 1.6 → 0.6
Time: 3.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.23303180311761695 \cdot 10^{88} \lor \neg \left(x \le 4.01567555002013953 \cdot 10^{-110}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.23303180311761695 \cdot 10^{88} \lor \neg \left(x \le 4.01567555002013953 \cdot 10^{-110}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r28404 = x;
        double r28405 = 4.0;
        double r28406 = r28404 + r28405;
        double r28407 = y;
        double r28408 = r28406 / r28407;
        double r28409 = r28404 / r28407;
        double r28410 = z;
        double r28411 = r28409 * r28410;
        double r28412 = r28408 - r28411;
        double r28413 = fabs(r28412);
        return r28413;
}
↓
double f(double x, double y, double z) {
        double r28414 = x;
        double r28415 = -3.233031803117617e+88;
        bool r28416 = r28414 <= r28415;
        double r28417 = 4.0156755500201395e-110;
        bool r28418 = r28414 <= r28417;
        double r28419 = !r28418;
        bool r28420 = r28416 || r28419;
        double r28421 = 4.0;
        double r28422 = r28414 + r28421;
        double r28423 = y;
        double r28424 = r28422 / r28423;
        double r28425 = z;
        double r28426 = r28425 / r28423;
        double r28427 = r28414 * r28426;
        double r28428 = r28424 - r28427;
        double r28429 = fabs(r28428);
        double r28430 = r28414 * r28425;
        double r28431 = 1.0;
        double r28432 = r28431 / r28423;
        double r28433 = r28430 * r28432;
        double r28434 = r28424 - r28433;
        double r28435 = fabs(r28434);
        double r28436 = r28420 ? r28429 : r28435;
        return r28436;
}

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 < -3.233031803117617e+88 or 4.0156755500201395e-110 < x

    1. Initial program 0.6

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

    3. Applied div-inv0.6

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

    4. Applied associate-*l*0.8

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

    5. Simplified0.7

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

    if -3.233031803117617e+88 < x < 4.0156755500201395e-110

    1. Initial program 2.4

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

    3. Applied div-inv2.4

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

    4. Applied associate-*l*5.2

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

    5. Simplified5.2

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

    7. Applied div-inv5.2

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

    8. Applied associate-*r*0.4

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot z\right) \cdot \frac{1}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.23303180311761695 \cdot 10^{88} \lor \neg \left(x \le 4.01567555002013953 \cdot 10^{-110}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))