Average Error: 1.6 → 0.6
Time: 12.1s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{4 + x}{y} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{4 + x}{y} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right|
double f(double x, double y, double z) {
        double r53197 = x;
        double r53198 = 4.0;
        double r53199 = r53197 + r53198;
        double r53200 = y;
        double r53201 = r53199 / r53200;
        double r53202 = r53197 / r53200;
        double r53203 = z;
        double r53204 = r53202 * r53203;
        double r53205 = r53201 - r53204;
        double r53206 = fabs(r53205);
        return r53206;
}
↓
double f(double x, double y, double z) {
        double r53207 = 4.0;
        double r53208 = x;
        double r53209 = r53207 + r53208;
        double r53210 = y;
        double r53211 = r53209 / r53210;
        double r53212 = cbrt(r53208);
        double r53213 = r53212 * r53212;
        double r53214 = cbrt(r53210);
        double r53215 = r53214 * r53214;
        double r53216 = r53213 / r53215;
        double r53217 = z;
        double r53218 = r53212 / r53214;
        double r53219 = r53217 * r53218;
        double r53220 = r53216 * r53219;
        double r53221 = r53211 - r53220;
        double r53222 = fabs(r53221);
        return r53222;
}

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

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

  4. Applied add-cube-cbrt1.9

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

  5. Applied times-frac2.0

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

  6. Applied associate-*l*0.6

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

  7. Simplified1.2

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

  9. Applied *-un-lft-identity1.2

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

  10. Applied cbrt-prod1.2

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

  11. Applied times-frac0.6

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

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