Average Error: 1.8 → 3.4
Time: 13.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|
double f(double x, double y, double z) {
        double r40739 = x;
        double r40740 = 4.0;
        double r40741 = r40739 + r40740;
        double r40742 = y;
        double r40743 = r40741 / r40742;
        double r40744 = r40739 / r40742;
        double r40745 = z;
        double r40746 = r40744 * r40745;
        double r40747 = r40743 - r40746;
        double r40748 = fabs(r40747);
        return r40748;
}
↓
double f(double x, double y, double z) {
        double r40749 = x;
        double r40750 = 4.0;
        double r40751 = r40749 + r40750;
        double r40752 = z;
        double r40753 = r40749 * r40752;
        double r40754 = r40751 - r40753;
        double r40755 = y;
        double r40756 = r40754 / r40755;
        double r40757 = fabs(r40756);
        return r40757;
}

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 < -7.629642207394974e+129 or 9.321223284732019e+58 < 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

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

    4. Applied add-cube-cbrt0.5

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

    5. Applied times-frac0.5

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

    6. Applied associate-*l*0.5

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

    8. Applied *-un-lft-identity0.5

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

    9. Applied associate-*l*0.5

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

    10. Simplified0.1

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

    11. Taylor expanded around 0 0.1

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

    if -7.629642207394974e+129 < x < 9.321223284732019e+58

    1. Initial program 2.4

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

    3. Applied associate-*l/0.6

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

    4. Applied sub-div0.6

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

  4. Final simplification3.4

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))