Average Error: 1.6 → 1.2
Time: 21.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{x + 4}{y} - \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\frac{\sqrt[3]{y}}{x}}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{x + 4}{y} - \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\frac{\sqrt[3]{y}}{x}}\right|
double f(double x, double y, double z) {
        double r72964 = x;
        double r72965 = 4.0;
        double r72966 = r72964 + r72965;
        double r72967 = y;
        double r72968 = r72966 / r72967;
        double r72969 = r72964 / r72967;
        double r72970 = z;
        double r72971 = r72969 * r72970;
        double r72972 = r72968 - r72971;
        double r72973 = fabs(r72972);
        return r72973;
}
↓
double f(double x, double y, double z) {
        double r72974 = x;
        double r72975 = 4.0;
        double r72976 = r72974 + r72975;
        double r72977 = y;
        double r72978 = r72976 / r72977;
        double r72979 = z;
        double r72980 = cbrt(r72979);
        double r72981 = r72980 * r72980;
        double r72982 = cbrt(r72977);
        double r72983 = r72982 * r72982;
        double r72984 = r72981 / r72983;
        double r72985 = r72982 / r72974;
        double r72986 = r72980 / r72985;
        double r72987 = r72984 * r72986;
        double r72988 = r72978 - r72987;
        double r72989 = fabs(r72988);
        return r72989;
}

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. Using strategy rm

  3. Applied add-cube-cbrt1.9

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

  4. Applied associate-*r*1.9

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

  5. Simplified1.9

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

  7. Applied pow11.9

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

  8. Applied pow11.9

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

  9. Applied pow11.9

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

  10. Applied pow11.9

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

  11. Applied pow-prod-down1.9

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

  12. Applied pow-prod-down1.9

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

  13. Applied pow-prod-down1.9

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

  14. Simplified1.8

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

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

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

  17. Applied add-cube-cbrt2.0

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

  18. Applied times-frac2.0

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

  19. Applied add-cube-cbrt2.1

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

  20. Applied times-frac1.2

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

  21. Simplified1.2

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

  22. Final simplification1.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))