Average Error: 1.6 → 0.8
Time: 19.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right) \cdot x\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \left(\left(\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right) \cdot x\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right|
double f(double x, double y, double z) {
        double r1684150 = x;
        double r1684151 = 4.0;
        double r1684152 = r1684150 + r1684151;
        double r1684153 = y;
        double r1684154 = r1684152 / r1684153;
        double r1684155 = r1684150 / r1684153;
        double r1684156 = z;
        double r1684157 = r1684155 * r1684156;
        double r1684158 = r1684154 - r1684157;
        double r1684159 = fabs(r1684158);
        return r1684159;
}
↓
double f(double x, double y, double z) {
        double r1684160 = 4.0;
        double r1684161 = y;
        double r1684162 = r1684160 / r1684161;
        double r1684163 = x;
        double r1684164 = r1684163 / r1684161;
        double r1684165 = r1684162 + r1684164;
        double r1684166 = z;
        double r1684167 = cbrt(r1684166);
        double r1684168 = cbrt(r1684161);
        double r1684169 = r1684167 / r1684168;
        double r1684170 = r1684169 * r1684169;
        double r1684171 = r1684170 * r1684163;
        double r1684172 = r1684171 * r1684169;
        double r1684173 = r1684165 - r1684172;
        double r1684174 = fabs(r1684173);
        return r1684174;
}

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 1.6

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

  2. Taylor expanded around -inf 1.6

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

  3. Simplified1.6

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

  5. Applied div-inv1.6

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

  6. Applied associate-*l*3.2

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

  7. Simplified3.2

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

  9. Applied add-cube-cbrt3.5

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

  10. Applied add-cube-cbrt3.5

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

  11. Applied times-frac3.5

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

  12. Applied associate-*r*0.8

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

  13. Simplified0.8

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

  14. Final simplification0.8

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))