Average Error: 0.1 → 0.2
Time: 4.9s
Precision: binary64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]
x \cdot \cos y - z \cdot \sin y
↓
\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y
(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z)
 :precision binary64
 (-
  (* (* x (pow (pow (cos y) 2.0) 0.3333333333333333)) (cbrt (cos y)))
  (* z (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) (x * ((double) pow(((double) pow(((double) cos(y)), 2.0)), 0.3333333333333333)))) * ((double) cbrt(((double) cos(y)))))) - ((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 Error: 0.1 bits

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

  3. Applied add-cube-cbrtError: 0.4 bits

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

  4. Applied associate-*r*Error: 0.4 bits

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

  6. Applied pow1/3Error: 16.5 bits

    \[\leadsto \left(x \cdot \left(\sqrt[3]{\cos y} \cdot \color{blue}{{\left(\cos y\right)}^{0.3333333333333333}}\right)\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  7. Applied pow1/3Error: 16.4 bits

    \[\leadsto \left(x \cdot \left(\color{blue}{{\left(\cos y\right)}^{0.3333333333333333}} \cdot {\left(\cos y\right)}^{0.3333333333333333}\right)\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  8. Applied pow-prod-downError: 0.2 bits

    \[\leadsto \left(x \cdot \color{blue}{{\left(\cos y \cdot \cos y\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  9. SimplifiedError: 0.2 bits

    \[\leadsto \left(x \cdot {\color{blue}{\left({\left(\cos y\right)}^{2}\right)}}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  10. Final simplificationError: 0.2 bits

    \[\leadsto \left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

Reproduce

herbie shell --seed 2020204 
(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))))