Average Error: 1.4 → 1.7
Time: 12.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.968897763889820057096245510304041678837 \cdot 10^{89}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.968897763889820057096245510304041678837 \cdot 10^{89}:\\
\;\;\;\;\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 r22011 = x;
        double r22012 = 4.0;
        double r22013 = r22011 + r22012;
        double r22014 = y;
        double r22015 = r22013 / r22014;
        double r22016 = r22011 / r22014;
        double r22017 = z;
        double r22018 = r22016 * r22017;
        double r22019 = r22015 - r22018;
        double r22020 = fabs(r22019);
        return r22020;
}
↓
double f(double x, double y, double z) {
        double r22021 = x;
        double r22022 = 1.96889776388982e+89;
        bool r22023 = r22021 <= r22022;
        double r22024 = 4.0;
        double r22025 = r22021 + r22024;
        double r22026 = y;
        double r22027 = r22025 / r22026;
        double r22028 = z;
        double r22029 = r22021 * r22028;
        double r22030 = r22029 / r22026;
        double r22031 = r22027 - r22030;
        double r22032 = fabs(r22031);
        double r22033 = r22028 / r22026;
        double r22034 = r22021 * r22033;
        double r22035 = r22027 - r22034;
        double r22036 = fabs(r22035);
        double r22037 = r22023 ? r22032 : r22036;
        return r22037;
}

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 2 regimes

  2. if x < 1.96889776388982e+89

    1. Initial program 1.6

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

    3. Applied *-un-lft-identity1.6

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

    5. Applied associate-*l/1.9

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

    6. Simplified1.9

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

    if 1.96889776388982e+89 < x

    1. Initial program 0.1

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

    3. Applied *-un-lft-identity0.1

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

    5. Applied div-inv0.1

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

    6. Applied associate-*l*0.1

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

    7. Simplified0.1

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.968897763889820057096245510304041678837 \cdot 10^{89}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))