Average Error: 1.5 → 0.4
Time: 15.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -16246640829.1726360321044921875:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1443179 = x;
        double r1443180 = 4.0;
        double r1443181 = r1443179 + r1443180;
        double r1443182 = y;
        double r1443183 = r1443181 / r1443182;
        double r1443184 = r1443179 / r1443182;
        double r1443185 = z;
        double r1443186 = r1443184 * r1443185;
        double r1443187 = r1443183 - r1443186;
        double r1443188 = fabs(r1443187);
        return r1443188;
}
↓
double f(double x, double y, double z) {
        double r1443189 = x;
        double r1443190 = -16246640829.172636;
        bool r1443191 = r1443189 <= r1443190;
        double r1443192 = 4.0;
        double r1443193 = y;
        double r1443194 = r1443192 / r1443193;
        double r1443195 = r1443189 / r1443193;
        double r1443196 = r1443194 + r1443195;
        double r1443197 = z;
        double r1443198 = r1443195 * r1443197;
        double r1443199 = r1443196 - r1443198;
        double r1443200 = fabs(r1443199);
        double r1443201 = 1.0296033599100243e-161;
        bool r1443202 = r1443189 <= r1443201;
        double r1443203 = r1443192 + r1443189;
        double r1443204 = r1443189 * r1443197;
        double r1443205 = r1443203 - r1443204;
        double r1443206 = r1443205 / r1443193;
        double r1443207 = fabs(r1443206);
        double r1443208 = r1443202 ? r1443207 : r1443200;
        double r1443209 = r1443191 ? r1443200 : r1443208;
        return r1443209;
}

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 < -16246640829.172636 or 1.0296033599100243e-161 < x

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    if -16246640829.172636 < x < 1.0296033599100243e-161

    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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -16246640829.1726360321044921875:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.029603359910024341294037093153671818016 \cdot 10^{-161}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))