Average Error: 0.1 → 0.3
Time: 4.8s
Precision: binary64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[\sqrt[3]{{\left(\cos y\right)}^{2}} \cdot \left(\sqrt[3]{\cos y} \cdot x\right) + z \cdot \sin y\]
x \cdot \cos y + z \cdot \sin y
↓
\sqrt[3]{{\left(\cos y\right)}^{2}} \cdot \left(\sqrt[3]{\cos y} \cdot x\right) + z \cdot \sin y
(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z)
 :precision binary64
 (+ (* (cbrt (pow (cos y) 2.0)) (* (cbrt (cos y)) x)) (* 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) cbrt(((double) pow(((double) cos(y)), 2.0)))) * ((double) (((double) cbrt(((double) cos(y)))) * x)))) + ((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 cbrt-unprodError: 0.3 bits

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

  7. SimplifiedError: 0.3 bits

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

  9. Applied pow1Error: 0.3 bits

    \[\leadsto \left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \color{blue}{{\left(\sqrt[3]{\cos y}\right)}^{1}} + z \cdot \sin y\]

  10. Applied pow1Error: 0.3 bits

    \[\leadsto \left(x \cdot \color{blue}{{\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{1}}\right) \cdot {\left(\sqrt[3]{\cos y}\right)}^{1} + z \cdot \sin y\]

  11. Applied pow1Error: 0.3 bits

    \[\leadsto \left(\color{blue}{{x}^{1}} \cdot {\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{1}\right) \cdot {\left(\sqrt[3]{\cos y}\right)}^{1} + z \cdot \sin y\]

  12. Applied pow-prod-downError: 0.3 bits

    \[\leadsto \color{blue}{{\left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right)}^{1}} \cdot {\left(\sqrt[3]{\cos y}\right)}^{1} + z \cdot \sin y\]

  13. Applied pow-prod-downError: 0.3 bits

    \[\leadsto \color{blue}{{\left(\left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \sqrt[3]{\cos y}\right)}^{1}} + z \cdot \sin y\]

  14. SimplifiedError: 0.3 bits

    \[\leadsto {\color{blue}{\left(\sqrt[3]{{\left(\cos y\right)}^{2}} \cdot \left(\sqrt[3]{\cos y} \cdot x\right)\right)}}^{1} + z \cdot \sin y\]

  15. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020204 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))