Average Error: 1.6 → 0.8
Time: 13.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \left(x \cdot \left(\frac{\sqrt[3]{z}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right)\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \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|
double f(double x, double y, double z) {
        double r1083065 = x;
        double r1083066 = 4.0;
        double r1083067 = r1083065 + r1083066;
        double r1083068 = y;
        double r1083069 = r1083067 / r1083068;
        double r1083070 = r1083065 / r1083068;
        double r1083071 = z;
        double r1083072 = r1083070 * r1083071;
        double r1083073 = r1083069 - r1083072;
        double r1083074 = fabs(r1083073);
        return r1083074;
}
↓
double f(double x, double y, double z) {
        double r1083075 = x;
        double r1083076 = y;
        double r1083077 = r1083075 / r1083076;
        double r1083078 = 4.0;
        double r1083079 = r1083078 / r1083076;
        double r1083080 = r1083077 + r1083079;
        double r1083081 = z;
        double r1083082 = cbrt(r1083081);
        double r1083083 = cbrt(r1083076);
        double r1083084 = r1083082 / r1083083;
        double r1083085 = r1083084 * r1083084;
        double r1083086 = r1083075 * r1083085;
        double r1083087 = r1083086 * r1083084;
        double r1083088 = r1083080 - r1083087;
        double r1083089 = fabs(r1083088);
        return r1083089;
}

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

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

  5. Applied div-inv1.6

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

  6. Applied associate-*l*3.2

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

  7. Simplified3.2

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

  9. Applied add-cube-cbrt3.5

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

  10. Applied add-cube-cbrt3.5

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

  11. Applied times-frac3.5

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

  12. Applied associate-*r*0.8

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

  13. Simplified0.8

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

  14. Final simplification0.8

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

Reproduce

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