Average Error: 0.1 → 0.4
Time: 5.3s
Precision: binary64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[\sqrt[3]{\cos y} \cdot \left(x \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos y}} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right) \cdot {\left({\left(\cos y\right)}^{4}\right)}^{0.1111111111111111}\right)\right) + z \cdot \sin y\]
x \cdot \cos y + z \cdot \sin y
↓
\sqrt[3]{\cos y} \cdot \left(x \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos y}} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right) \cdot {\left({\left(\cos y\right)}^{4}\right)}^{0.1111111111111111}\right)\right) + z \cdot \sin y
double code(double x, double y, double z) {
	return ((double) (((double) (x * ((double) cos(y)))) + ((double) (z * ((double) sin(y))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (((double) (((double) cbrt(((double) cos(y)))) * ((double) (x * ((double) (((double) (((double) cbrt(((double) cbrt(((double) cos(y)))))) * ((double) cbrt(((double) cbrt(((double) cos(y)))))))) * ((double) pow(((double) pow(((double) cos(y)), 4.0)), 0.1111111111111111)))))))) + ((double) (z * ((double) 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 add-cube-cbrt0.4

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

  7. Applied cbrt-prod0.4

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

  8. Applied associate-*l*0.4

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

  9. Simplified0.5

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt[3]{\cos y}}\right)}^{4}}\right)\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]

  10. Taylor expanded around inf 0.3

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}} \cdot \color{blue}{{\left({\left(\cos y\right)}^{4}\right)}^{0.1111111111111111}}\right)\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]
  11. Using strategy rm

  12. Applied cbrt-prod0.4

    \[\leadsto \left(x \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{\cos y}} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right)} \cdot {\left({\left(\cos y\right)}^{4}\right)}^{0.1111111111111111}\right)\right) \cdot \sqrt[3]{\cos y} + z \cdot \sin y\]

  13. Final simplification0.4

    \[\leadsto \sqrt[3]{\cos y} \cdot \left(x \cdot \left(\left(\sqrt[3]{\sqrt[3]{\cos y}} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right) \cdot {\left({\left(\cos y\right)}^{4}\right)}^{0.1111111111111111}\right)\right) + z \cdot \sin y\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))