Average Error: 1.5 → 0.1
Time: 18.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 3.057947311561769 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
double f(double x, double y, double z) {
        double r1843494 = x;
        double r1843495 = 4.0;
        double r1843496 = r1843494 + r1843495;
        double r1843497 = y;
        double r1843498 = r1843496 / r1843497;
        double r1843499 = r1843494 / r1843497;
        double r1843500 = z;
        double r1843501 = r1843499 * r1843500;
        double r1843502 = r1843498 - r1843501;
        double r1843503 = fabs(r1843502);
        return r1843503;
}
↓
double f(double x, double y, double z) {
        double r1843504 = 4.0;
        double r1843505 = x;
        double r1843506 = r1843504 + r1843505;
        double r1843507 = y;
        double r1843508 = r1843506 / r1843507;
        double r1843509 = r1843505 / r1843507;
        double r1843510 = z;
        double r1843511 = r1843509 * r1843510;
        double r1843512 = r1843508 - r1843511;
        double r1843513 = fabs(r1843512);
        double r1843514 = 3.057947311561769e-53;
        bool r1843515 = r1843513 <= r1843514;
        double r1843516 = r1843505 * r1843510;
        double r1843517 = 1.0;
        double r1843518 = r1843517 / r1843507;
        double r1843519 = r1843516 * r1843518;
        double r1843520 = r1843508 - r1843519;
        double r1843521 = fabs(r1843520);
        double r1843522 = r1843515 ? r1843521 : r1843513;
        return r1843522;
}


\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 3.057947311561769 \cdot 10^{-53}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\

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

\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if (fabs (- (/ (+ x 4) y) (* (/ x y) z))) < 3.057947311561769e-53

    1. Initial program 4.7

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

    3. Applied *-un-lft-identity4.7

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

    4. Applied add-cube-cbrt4.8

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

    5. Applied times-frac4.8

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

    6. Applied associate-*l*1.6

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

    7. Simplified1.6

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

    8. Taylor expanded around 0 0.1

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

    10. Applied div-inv0.1

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

    if 3.057947311561769e-53 < (fabs (- (/ (+ x 4) y) (* (/ x y) z)))

    1. Initial program 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 3.057947311561769 \cdot 10^{-53}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))