Average Error: 1.6 → 0.5
Time: 3.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -71921344804687528:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 9.36037578809800899 \cdot 10^{-141}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x + 4, \frac{1}{y}, -\frac{x}{y} \cdot z\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -71921344804687528:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 9.36037578809800899 \cdot 10^{-141}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r35623 = x;
        double r35624 = 4.0;
        double r35625 = r35623 + r35624;
        double r35626 = y;
        double r35627 = r35625 / r35626;
        double r35628 = r35623 / r35626;
        double r35629 = z;
        double r35630 = r35628 * r35629;
        double r35631 = r35627 - r35630;
        double r35632 = fabs(r35631);
        return r35632;
}
↓
double f(double x, double y, double z) {
        double r35633 = x;
        double r35634 = -7.192134480468753e+16;
        bool r35635 = r35633 <= r35634;
        double r35636 = 4.0;
        double r35637 = r35633 + r35636;
        double r35638 = y;
        double r35639 = r35637 / r35638;
        double r35640 = z;
        double r35641 = r35640 / r35638;
        double r35642 = r35633 * r35641;
        double r35643 = r35639 - r35642;
        double r35644 = fabs(r35643);
        double r35645 = 9.360375788098009e-141;
        bool r35646 = r35633 <= r35645;
        double r35647 = r35633 * r35640;
        double r35648 = r35637 - r35647;
        double r35649 = r35648 / r35638;
        double r35650 = fabs(r35649);
        double r35651 = 1.0;
        double r35652 = r35651 / r35638;
        double r35653 = r35633 / r35638;
        double r35654 = r35653 * r35640;
        double r35655 = -r35654;
        double r35656 = fma(r35637, r35652, r35655);
        double r35657 = fabs(r35656);
        double r35658 = r35646 ? r35650 : r35657;
        double r35659 = r35635 ? r35644 : r35658;
        return r35659;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if x < -7.192134480468753e+16

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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. Simplified0.1

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

    if -7.192134480468753e+16 < x < 9.360375788098009e-141

    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

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

    if 9.360375788098009e-141 < x

    1. Initial program 1.3

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

    3. Applied div-inv1.3

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

    4. Applied fma-neg1.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -71921344804687528:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 9.36037578809800899 \cdot 10^{-141}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(x + 4, \frac{1}{y}, -\frac{x}{y} \cdot z\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))