Average Error: 1.6 → 0.1
Time: 21.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9993061933414586249117696 \lor \neg \left(x \le 1.755705061898593338352186152639686699093 \cdot 10^{-16}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -9993061933414586249117696 \lor \neg \left(x \le 1.755705061898593338352186152639686699093 \cdot 10^{-16}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r46737 = x;
        double r46738 = 4.0;
        double r46739 = r46737 + r46738;
        double r46740 = y;
        double r46741 = r46739 / r46740;
        double r46742 = r46737 / r46740;
        double r46743 = z;
        double r46744 = r46742 * r46743;
        double r46745 = r46741 - r46744;
        double r46746 = fabs(r46745);
        return r46746;
}
↓
double f(double x, double y, double z) {
        double r46747 = x;
        double r46748 = -9.993061933414586e+24;
        bool r46749 = r46747 <= r46748;
        double r46750 = 1.7557050618985933e-16;
        bool r46751 = r46747 <= r46750;
        double r46752 = !r46751;
        bool r46753 = r46749 || r46752;
        double r46754 = 4.0;
        double r46755 = r46747 + r46754;
        double r46756 = y;
        double r46757 = r46755 / r46756;
        double r46758 = z;
        double r46759 = r46758 / r46756;
        double r46760 = r46747 * r46759;
        double r46761 = r46757 - r46760;
        double r46762 = fabs(r46761);
        double r46763 = r46747 * r46758;
        double r46764 = r46763 / r46756;
        double r46765 = r46757 - r46764;
        double r46766 = fabs(r46765);
        double r46767 = r46753 ? r46762 : r46766;
        return r46767;
}

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 < -9.993061933414586e+24 or 1.7557050618985933e-16 < x

    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 -9.993061933414586e+24 < x < 1.7557050618985933e-16

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9993061933414586249117696 \lor \neg \left(x \le 1.755705061898593338352186152639686699093 \cdot 10^{-16}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))