Average Error: 0.1 → 0.2
Time: 5.0s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[x \cdot \sqrt[3]{{\left(\cos y\right)}^{3}} - z \cdot \sin y\]
x \cdot \cos y - z \cdot \sin y
↓
x \cdot \sqrt[3]{{\left(\cos y\right)}^{3}} - z \cdot \sin y
double code(double x, double y, double z) {
	return ((x * cos(y)) - (z * sin(y)));
}
↓
double code(double x, double y, double z) {
	return ((x * cbrt(pow(cos(y), 3.0))) - (z * sin(y)));
}

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 0.1

    \[x \cdot \cos y - z \cdot \sin y\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.4

    \[\leadsto x \cdot \color{blue}{\left(\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) \cdot \sqrt[3]{\cos y}\right)} - z \cdot \sin y\]

  4. Applied associate-*r*0.4

    \[\leadsto \color{blue}{\left(x \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}} - z \cdot \sin y\]
  5. Using strategy rm

  6. Applied pow1/316.6

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

  7. Applied pow1/316.6

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

  8. Applied pow-prod-down0.2

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

  9. Simplified0.2

    \[\leadsto \left(x \cdot {\color{blue}{\left({\left(\cos y\right)}^{2}\right)}}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]
  10. Using strategy rm

  11. Applied associate-*l*0.2

    \[\leadsto \color{blue}{x \cdot \left({\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\cos y}\right)} - z \cdot \sin y\]
  12. Using strategy rm

  13. Applied unpow1/30.3

    \[\leadsto x \cdot \left(\color{blue}{\sqrt[3]{{\left(\cos y\right)}^{2}}} \cdot \sqrt[3]{\cos y}\right) - z \cdot \sin y\]

  14. Applied cbrt-unprod0.2

    \[\leadsto x \cdot \color{blue}{\sqrt[3]{{\left(\cos y\right)}^{2} \cdot \cos y}} - z \cdot \sin y\]

  15. Simplified0.2

    \[\leadsto x \cdot \sqrt[3]{\color{blue}{{\left(\cos y\right)}^{3}}} - z \cdot \sin y\]

  16. Final simplification0.2

    \[\leadsto x \cdot \sqrt[3]{{\left(\cos y\right)}^{3}} - z \cdot \sin y\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))