Average Error: 0.1 → 0.2
Time: 5.2s
Precision: 64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[\mathsf{fma}\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}}, \log \left(e^{\sqrt[3]{\cos y}}\right), z \cdot \sin y\right)\]
x \cdot \cos y + z \cdot \sin y
↓
\mathsf{fma}\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}}, \log \left(e^{\sqrt[3]{\cos y}}\right), z \cdot \sin y\right)
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) fma(((double) (x * ((double) pow(((double) pow(((double) cos(y)), 2.0)), 0.3333333333333333)))), ((double) log(((double) exp(((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 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. Applied fma-def0.4

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

  7. Applied pow1/315.6

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

  8. Applied pow1/315.6

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

  9. Applied pow-prod-down0.2

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

  10. Simplified0.2

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

  12. Applied add-log-exp0.2

    \[\leadsto \mathsf{fma}\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}}, \color{blue}{\log \left(e^{\sqrt[3]{\cos y}}\right)}, z \cdot \sin y\right)\]

  13. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}}, \log \left(e^{\sqrt[3]{\cos y}}\right), z \cdot \sin y\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))