Average Error: 1.7 → 0.6
Time: 13.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|
double f(double x, double y, double z) {
        double r1299064 = x;
        double r1299065 = 4.0;
        double r1299066 = r1299064 + r1299065;
        double r1299067 = y;
        double r1299068 = r1299066 / r1299067;
        double r1299069 = r1299064 / r1299067;
        double r1299070 = z;
        double r1299071 = r1299069 * r1299070;
        double r1299072 = r1299068 - r1299071;
        double r1299073 = fabs(r1299072);
        return r1299073;
}
↓
double f(double x, double y, double z) {
        double r1299074 = x;
        double r1299075 = y;
        double r1299076 = r1299074 / r1299075;
        double r1299077 = 4.0;
        double r1299078 = r1299077 / r1299075;
        double r1299079 = r1299076 + r1299078;
        double r1299080 = cbrt(r1299074);
        double r1299081 = r1299080 * r1299080;
        double r1299082 = cbrt(r1299075);
        double r1299083 = r1299082 * r1299082;
        double r1299084 = r1299081 / r1299083;
        double r1299085 = r1299080 / r1299082;
        double r1299086 = z;
        double r1299087 = r1299085 * r1299086;
        double r1299088 = r1299084 * r1299087;
        double r1299089 = r1299079 - r1299088;
        double r1299090 = fabs(r1299089);
        return r1299090;
}

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. Initial program 1.7

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

  2. Taylor expanded around 0 1.7

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

  3. Simplified1.7

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

  5. Applied add-cube-cbrt2.0

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

  6. Applied add-cube-cbrt2.0

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

  7. Applied times-frac2.0

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

  8. Applied associate-*l*0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))