Average Error: 1.5 → 0.3
Time: 10.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.0025647343369246 \cdot 10^{88}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.38362149527649136 \cdot 10^{44}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.0025647343369246 \cdot 10^{88}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 2.38362149527649136 \cdot 10^{44}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r26750 = x;
        double r26751 = 4.0;
        double r26752 = r26750 + r26751;
        double r26753 = y;
        double r26754 = r26752 / r26753;
        double r26755 = r26750 / r26753;
        double r26756 = z;
        double r26757 = r26755 * r26756;
        double r26758 = r26754 - r26757;
        double r26759 = fabs(r26758);
        return r26759;
}
↓
double f(double x, double y, double z) {
        double r26760 = x;
        double r26761 = -4.0025647343369246e+88;
        bool r26762 = r26760 <= r26761;
        double r26763 = 4.0;
        double r26764 = y;
        double r26765 = r26763 / r26764;
        double r26766 = r26760 / r26764;
        double r26767 = r26765 + r26766;
        double r26768 = z;
        double r26769 = r26766 * r26768;
        double r26770 = r26767 - r26769;
        double r26771 = fabs(r26770);
        double r26772 = 2.3836214952764914e+44;
        bool r26773 = r26760 <= r26772;
        double r26774 = r26760 + r26763;
        double r26775 = r26760 * r26768;
        double r26776 = r26774 - r26775;
        double r26777 = r26776 / r26764;
        double r26778 = fabs(r26777);
        double r26779 = r26768 / r26764;
        double r26780 = r26760 * r26779;
        double r26781 = r26767 - r26780;
        double r26782 = fabs(r26781);
        double r26783 = r26773 ? r26778 : r26782;
        double r26784 = r26762 ? r26771 : r26783;
        return r26784;
}

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 3 regimes

  2. if x < -4.0025647343369246e+88

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    if -4.0025647343369246e+88 < x < 2.3836214952764914e+44

    1. Initial program 2.0

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

    3. Applied associate-*l/0.3

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

    4. Applied sub-div0.3

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

    if 2.3836214952764914e+44 < x

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied div-inv0.2

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

    6. Applied associate-*l*0.2

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

    7. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.0025647343369246 \cdot 10^{88}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.38362149527649136 \cdot 10^{44}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))