Average Error: 1.7 → 0.8
Time: 16.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.123189090912366160804526572354602418751 \cdot 10^{161}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;y \le 5.600885815035954071116474607611908162649 \cdot 10^{83}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -6.123189090912366160804526572354602418751 \cdot 10^{161}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\

\mathbf{elif}\;y \le 5.600885815035954071116474607611908162649 \cdot 10^{83}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - z \cdot x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1920653 = x;
        double r1920654 = 4.0;
        double r1920655 = r1920653 + r1920654;
        double r1920656 = y;
        double r1920657 = r1920655 / r1920656;
        double r1920658 = r1920653 / r1920656;
        double r1920659 = z;
        double r1920660 = r1920658 * r1920659;
        double r1920661 = r1920657 - r1920660;
        double r1920662 = fabs(r1920661);
        return r1920662;
}
↓
double f(double x, double y, double z) {
        double r1920663 = y;
        double r1920664 = -6.123189090912366e+161;
        bool r1920665 = r1920663 <= r1920664;
        double r1920666 = x;
        double r1920667 = 4.0;
        double r1920668 = r1920666 + r1920667;
        double r1920669 = r1920668 / r1920663;
        double r1920670 = z;
        double r1920671 = r1920663 / r1920670;
        double r1920672 = r1920666 / r1920671;
        double r1920673 = r1920669 - r1920672;
        double r1920674 = fabs(r1920673);
        double r1920675 = 5.600885815035954e+83;
        bool r1920676 = r1920663 <= r1920675;
        double r1920677 = r1920670 * r1920666;
        double r1920678 = r1920668 - r1920677;
        double r1920679 = r1920678 / r1920663;
        double r1920680 = fabs(r1920679);
        double r1920681 = r1920676 ? r1920680 : r1920674;
        double r1920682 = r1920665 ? r1920674 : r1920681;
        return r1920682;
}

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 y < -6.123189090912366e+161 or 5.600885815035954e+83 < y

    1. Initial program 4.0

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

    3. Applied *-un-lft-identity4.0

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

    4. Applied add-cube-cbrt4.4

      \[\leadsto \left|\frac{x + 4}{y} - \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot y} \cdot z\right|\]

    5. Applied times-frac4.4

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{y}\right)} \cdot z\right|\]

    6. Applied associate-*l*1.7

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \left(\frac{\sqrt[3]{x}}{y} \cdot z\right)}\right|\]
    7. Using strategy rm

    8. Applied *-un-lft-identity1.7

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(1 \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)} \cdot \left(\frac{\sqrt[3]{x}}{y} \cdot z\right)\right|\]

    9. Applied associate-*l*1.7

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{1 \cdot \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \left(\frac{\sqrt[3]{x}}{y} \cdot z\right)\right)}\right|\]

    10. Simplified0.1

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

    if -6.123189090912366e+161 < y < 5.600885815035954e+83

    1. Initial program 0.3

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

    3. Applied associate-*l/1.2

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

    4. Applied sub-div1.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6.123189090912366160804526572354602418751 \cdot 10^{161}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \mathbf{elif}\;y \le 5.600885815035954071116474607611908162649 \cdot 10^{83}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))