Average Error: 0.1 → 0.5
Time: 4.9s
Precision: binary64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[x \cdot \cos y + \sqrt[3]{z \cdot \sin y} \cdot \left(\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sin y}\right)\]
x \cdot \cos y + z \cdot \sin y
↓
x \cdot \cos y + \sqrt[3]{z \cdot \sin y} \cdot \left(\left(\sqrt[3]{z \cdot \sin y} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{\sin y}\right)
(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z)
 :precision binary64
 (+
  (* x (cos y))
  (*
   (cbrt (* z (sin y)))
   (* (* (cbrt (* z (sin y))) (cbrt z)) (cbrt (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) (x * ((double) cos(y)))) + ((double) (((double) cbrt(((double) (z * ((double) sin(y)))))) * ((double) (((double) (((double) cbrt(((double) (z * ((double) sin(y)))))) * ((double) cbrt(z)))) * ((double) cbrt(((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: 0.6 bits

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

  5. Applied cbrt-prodError: 0.5 bits

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

  6. Applied associate-*r*Error: 0.5 bits

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

  7. Final simplificationError: 0.5 bits

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

Reproduce

herbie shell --seed 2020203 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))