Average Error: 1.6 → 0.8
Time: 3.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 249476286.057907402515411376953125:\\ \;\;\;\;\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y} \cdot \left(1 - z\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le 249476286.057907402515411376953125:\\
\;\;\;\;\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y} \cdot \left(1 - z\right)\right)\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 r28024 = x;
        double r28025 = 4.0;
        double r28026 = r28024 + r28025;
        double r28027 = y;
        double r28028 = r28026 / r28027;
        double r28029 = r28024 / r28027;
        double r28030 = z;
        double r28031 = r28029 * r28030;
        double r28032 = r28028 - r28031;
        double r28033 = fabs(r28032);
        return r28033;
}
↓
double f(double x, double y, double z) {
        double r28034 = y;
        double r28035 = 249476286.0579074;
        bool r28036 = r28034 <= r28035;
        double r28037 = 4.0;
        double r28038 = 1.0;
        double r28039 = r28038 / r28034;
        double r28040 = x;
        double r28041 = r28040 / r28034;
        double r28042 = z;
        double r28043 = r28038 - r28042;
        double r28044 = r28041 * r28043;
        double r28045 = fma(r28037, r28039, r28044);
        double r28046 = fabs(r28045);
        double r28047 = r28040 + r28037;
        double r28048 = r28047 / r28034;
        double r28049 = r28042 / r28034;
        double r28050 = r28040 * r28049;
        double r28051 = r28048 - r28050;
        double r28052 = fabs(r28051);
        double r28053 = r28036 ? r28046 : r28052;
        return r28053;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if y < 249476286.0579074

    1. Initial program 1.1

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

    2. Taylor expanded around 0 2.3

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

    3. Simplified1.1

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

    if 249476286.0579074 < y

    1. Initial program 2.6

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

    3. Applied div-inv2.6

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

    4. Applied associate-*l*0.1

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

    5. Simplified0.1

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 249476286.057907402515411376953125:\\ \;\;\;\;\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y} \cdot \left(1 - z\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))