Average Error: 1.6 → 0.3
Time: 15.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.715123584991496019376606750892028249593 \cdot 10^{82} \lor \neg \left(y \le 4.614698310712485342741186148571646300614 \cdot 10^{-65}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.715123584991496019376606750892028249593 \cdot 10^{82} \lor \neg \left(y \le 4.614698310712485342741186148571646300614 \cdot 10^{-65}\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 r41527 = x;
        double r41528 = 4.0;
        double r41529 = r41527 + r41528;
        double r41530 = y;
        double r41531 = r41529 / r41530;
        double r41532 = r41527 / r41530;
        double r41533 = z;
        double r41534 = r41532 * r41533;
        double r41535 = r41531 - r41534;
        double r41536 = fabs(r41535);
        return r41536;
}
↓
double f(double x, double y, double z) {
        double r41537 = y;
        double r41538 = -2.715123584991496e+82;
        bool r41539 = r41537 <= r41538;
        double r41540 = 4.614698310712485e-65;
        bool r41541 = r41537 <= r41540;
        double r41542 = !r41541;
        bool r41543 = r41539 || r41542;
        double r41544 = x;
        double r41545 = 4.0;
        double r41546 = r41544 + r41545;
        double r41547 = r41546 / r41537;
        double r41548 = z;
        double r41549 = r41548 / r41537;
        double r41550 = r41544 * r41549;
        double r41551 = r41547 - r41550;
        double r41552 = fabs(r41551);
        double r41553 = r41544 * r41548;
        double r41554 = r41546 - r41553;
        double r41555 = r41554 / r41537;
        double r41556 = fabs(r41555);
        double r41557 = r41543 ? r41552 : r41556;
        return r41557;
}

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 < -2.715123584991496e+82 or 4.614698310712485e-65 < y

    1. Initial program 2.7

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

    3. Applied div-inv2.7

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

    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 -2.715123584991496e+82 < y < 4.614698310712485e-65

    1. Initial program 0.2

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.715123584991496019376606750892028249593 \cdot 10^{82} \lor \neg \left(y \le 4.614698310712485342741186148571646300614 \cdot 10^{-65}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))