Average Error: 0.1 → 0.2
Time: 5.6s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\mathsf{fma}\left(x \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos y\right)}^{2}\right)\right)\right)}^{\frac{1}{3}}, \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right), -\sin y \cdot z\right)\]
x \cdot \cos y - z \cdot \sin y
↓
\mathsf{fma}\left(x \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos y\right)}^{2}\right)\right)\right)}^{\frac{1}{3}}, \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right), -\sin y \cdot z\right)
double code(double x, double y, double z) {
	return ((x * cos(y)) - (z * sin(y)));
}
↓
double code(double x, double y, double z) {
	return fma((x * pow(expm1(log1p(pow(cos(y), 2.0))), 0.3333333333333333)), log1p(expm1(cbrt(cos(y)))), -(sin(y) * z));
}

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-neg0.4

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

  6. Simplified0.4

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

  8. Applied pow1/315.4

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

  9. Applied pow1/315.3

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

  10. Applied pow-prod-down0.2

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

  11. Simplified0.2

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

  13. Applied log1p-expm1-u0.2

    \[\leadsto \mathsf{fma}\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{3}}, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)}, -\sin y \cdot z\right)\]
  14. Using strategy rm

  15. Applied expm1-log1p-u0.2

    \[\leadsto \mathsf{fma}\left(x \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos y\right)}^{2}\right)\right)\right)}}^{\frac{1}{3}}, \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right), -\sin y \cdot z\right)\]

  16. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(x \cdot {\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\cos y\right)}^{2}\right)\right)\right)}^{\frac{1}{3}}, \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right), -\sin y \cdot z\right)\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))))