Average Error: 1.7 → 1.0
Time: 3.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \left(1 - z\right)\right) + 4 \cdot \frac{1}{y}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\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 \left(1 - z\right)\right) + 4 \cdot \frac{1}{y}\right|
double f(double x, double y, double z) {
        double r29952 = x;
        double r29953 = 4.0;
        double r29954 = r29952 + r29953;
        double r29955 = y;
        double r29956 = r29954 / r29955;
        double r29957 = r29952 / r29955;
        double r29958 = z;
        double r29959 = r29957 * r29958;
        double r29960 = r29956 - r29959;
        double r29961 = fabs(r29960);
        return r29961;
}
↓
double f(double x, double y, double z) {
        double r29962 = x;
        double r29963 = cbrt(r29962);
        double r29964 = r29963 * r29963;
        double r29965 = y;
        double r29966 = cbrt(r29965);
        double r29967 = r29966 * r29966;
        double r29968 = r29964 / r29967;
        double r29969 = r29963 / r29966;
        double r29970 = 1.0;
        double r29971 = z;
        double r29972 = r29970 - r29971;
        double r29973 = r29969 * r29972;
        double r29974 = r29968 * r29973;
        double r29975 = 4.0;
        double r29976 = r29970 / r29965;
        double r29977 = r29975 * r29976;
        double r29978 = r29974 + r29977;
        double r29979 = fabs(r29978);
        return r29979;
}

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 3.5

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

  3. Simplified1.7

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

  5. Applied add-cube-cbrt2.3

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

  6. Applied add-cube-cbrt2.5

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

  7. Applied times-frac2.5

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

  8. Applied associate-*l*1.0

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

  9. Final simplification1.0

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

Reproduce

herbie shell --seed 2020024 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))