Average Error: 1.5 → 0.8
Time: 23.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -30231216599687348029067407439959515805520000:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -30231216599687348029067407439959515805520000:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1363704 = x;
        double r1363705 = 4.0;
        double r1363706 = r1363704 + r1363705;
        double r1363707 = y;
        double r1363708 = r1363706 / r1363707;
        double r1363709 = r1363704 / r1363707;
        double r1363710 = z;
        double r1363711 = r1363709 * r1363710;
        double r1363712 = r1363708 - r1363711;
        double r1363713 = fabs(r1363712);
        return r1363713;
}
↓
double f(double x, double y, double z) {
        double r1363714 = y;
        double r1363715 = -3.023121659968735e+43;
        bool r1363716 = r1363714 <= r1363715;
        double r1363717 = 4.0;
        double r1363718 = x;
        double r1363719 = r1363717 + r1363718;
        double r1363720 = r1363719 / r1363714;
        double r1363721 = z;
        double r1363722 = r1363714 / r1363721;
        double r1363723 = r1363718 / r1363722;
        double r1363724 = r1363720 - r1363723;
        double r1363725 = fabs(r1363724);
        double r1363726 = r1363718 / r1363714;
        double r1363727 = 1.0;
        double r1363728 = r1363727 - r1363721;
        double r1363729 = r1363717 / r1363714;
        double r1363730 = fma(r1363726, r1363728, r1363729);
        double r1363731 = fabs(r1363730);
        double r1363732 = r1363716 ? r1363725 : r1363731;
        return r1363732;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if y < -3.023121659968735e+43

    1. Initial program 3.0

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

    3. Applied *-un-lft-identity3.0

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

    4. Applied add-cube-cbrt3.3

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

    5. Applied times-frac3.3

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

    6. Applied associate-*l*1.4

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

    8. Applied associate-*l/1.4

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

    9. Simplified0.1

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

    if -3.023121659968735e+43 < y

    1. Initial program 1.0

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

    2. Taylor expanded around 0 2.2

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

    3. Simplified1.0

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -30231216599687348029067407439959515805520000:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))