Average Error: 1.5 → 0.1
Time: 3.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.5424594366857613 \cdot 10^{29}:\\ \;\;\;\;\left|\left(\left(4 \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right)\right) \cdot \sqrt[3]{\frac{1}{y}} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.1896364681112564 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.5424594366857613 \cdot 10^{29}:\\
\;\;\;\;\left|\left(\left(4 \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right)\right) \cdot \sqrt[3]{\frac{1}{y}} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 2.1896364681112564 \cdot 10^{-12}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r29357 = x;
        double r29358 = 4.0;
        double r29359 = r29357 + r29358;
        double r29360 = y;
        double r29361 = r29359 / r29360;
        double r29362 = r29357 / r29360;
        double r29363 = z;
        double r29364 = r29362 * r29363;
        double r29365 = r29361 - r29364;
        double r29366 = fabs(r29365);
        return r29366;
}
↓
double f(double x, double y, double z) {
        double r29367 = x;
        double r29368 = -1.5424594366857613e+29;
        bool r29369 = r29367 <= r29368;
        double r29370 = 4.0;
        double r29371 = 1.0;
        double r29372 = y;
        double r29373 = r29371 / r29372;
        double r29374 = cbrt(r29373);
        double r29375 = r29374 * r29374;
        double r29376 = r29370 * r29375;
        double r29377 = r29376 * r29374;
        double r29378 = r29367 / r29372;
        double r29379 = r29377 + r29378;
        double r29380 = z;
        double r29381 = r29378 * r29380;
        double r29382 = r29379 - r29381;
        double r29383 = fabs(r29382);
        double r29384 = 2.1896364681112564e-12;
        bool r29385 = r29367 <= r29384;
        double r29386 = r29367 + r29370;
        double r29387 = r29386 / r29372;
        double r29388 = r29367 * r29380;
        double r29389 = r29388 / r29372;
        double r29390 = r29387 - r29389;
        double r29391 = fabs(r29390);
        double r29392 = r29380 / r29372;
        double r29393 = r29367 * r29392;
        double r29394 = r29387 - r29393;
        double r29395 = fabs(r29394);
        double r29396 = r29385 ? r29391 : r29395;
        double r29397 = r29369 ? r29383 : r29396;
        return r29397;
}

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 3 regimes

  2. if x < -1.5424594366857613e+29

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    4. Applied add-cube-cbrt0.1

      \[\leadsto \left|\left(4 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right) \cdot \sqrt[3]{\frac{1}{y}}\right)} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]

    5. Applied associate-*r*0.1

      \[\leadsto \left|\left(\color{blue}{\left(4 \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right)\right) \cdot \sqrt[3]{\frac{1}{y}}} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]

    if -1.5424594366857613e+29 < x < 2.1896364681112564e-12

    1. Initial program 2.5

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

    3. Applied div-inv2.5

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

    4. Applied associate-*l*5.8

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

    5. Simplified5.7

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

    7. Applied associate-*r/0.1

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

    if 2.1896364681112564e-12 < 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*0.2

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

    5. Simplified0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.5424594366857613 \cdot 10^{29}:\\ \;\;\;\;\left|\left(\left(4 \cdot \left(\sqrt[3]{\frac{1}{y}} \cdot \sqrt[3]{\frac{1}{y}}\right)\right) \cdot \sqrt[3]{\frac{1}{y}} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.1896364681112564 \cdot 10^{-12}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))