Average Error: 1.4 → 0.6
Time: 21.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(1, \frac{x + 4}{y}, -\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right) \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(-\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right) + \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(1, \frac{x + 4}{y}, -\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right) \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\left(-\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right) + \frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right|
double code(double x, double y, double z) {
	return fabs((((x + 4.0) / y) - ((x / y) * z)));
}
↓
double code(double x, double y, double z) {
	return fabs((fma(1.0, ((x + 4.0) / y), -(((cbrt(x) / cbrt(y)) * z) * ((cbrt(x) * cbrt(x)) / (cbrt(y) * cbrt(y))))) + (((cbrt(x) * cbrt(x)) / (cbrt(y) * cbrt(y))) * (-((cbrt(x) / cbrt(y)) * z) + ((cbrt(x) / cbrt(y)) * z)))));
}

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 1.4

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.7

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

  4. Applied add-cube-cbrt1.8

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

  5. Applied times-frac1.8

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

  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. Applied add-sqr-sqrt31.8

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

  8. Applied prod-diff31.8

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

  9. Simplified0.6

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

  10. Simplified0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))