Average Error: 1.5 → 0.4
Time: 22.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -16246640829.1726360321044921875:\\
\;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\

\mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r2305414 = x;
        double r2305415 = 4.0;
        double r2305416 = r2305414 + r2305415;
        double r2305417 = y;
        double r2305418 = r2305416 / r2305417;
        double r2305419 = r2305414 / r2305417;
        double r2305420 = z;
        double r2305421 = r2305419 * r2305420;
        double r2305422 = r2305418 - r2305421;
        double r2305423 = fabs(r2305422);
        return r2305423;
}
↓
double f(double x, double y, double z) {
        double r2305424 = x;
        double r2305425 = -16246640829.172636;
        bool r2305426 = r2305424 <= r2305425;
        double r2305427 = 1.0;
        double r2305428 = z;
        double r2305429 = r2305427 - r2305428;
        double r2305430 = y;
        double r2305431 = r2305424 / r2305430;
        double r2305432 = r2305429 * r2305431;
        double r2305433 = 4.0;
        double r2305434 = r2305433 / r2305430;
        double r2305435 = r2305432 + r2305434;
        double r2305436 = fabs(r2305435);
        double r2305437 = 1.0296033599100243e-161;
        bool r2305438 = r2305424 <= r2305437;
        double r2305439 = r2305433 + r2305424;
        double r2305440 = r2305424 * r2305428;
        double r2305441 = r2305439 - r2305440;
        double r2305442 = r2305441 / r2305430;
        double r2305443 = fabs(r2305442);
        double r2305444 = r2305438 ? r2305443 : r2305436;
        double r2305445 = r2305426 ? r2305436 : r2305444;
        return r2305445;
}

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 < -16246640829.172636 or 1.0296033599100243e-161 < x

    1. Initial program 0.7

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

    2. Taylor expanded around 0 6.2

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

    3. Simplified0.7

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

    if -16246640829.172636 < x < 1.0296033599100243e-161

    1. Initial program 2.5

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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(1 - z\right) \cdot \frac{x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))