Average Error: 0.1 → 0.3
Time: 5.6s
Precision: 64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[\mathsf{fma}\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)\right), \sqrt[3]{\cos y}, z \cdot \sin y\right)\]
x \cdot \cos y + z \cdot \sin y
↓
\mathsf{fma}\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\left(\cos y\right)}^{2}}\right)\right), \sqrt[3]{\cos y}, z \cdot \sin y\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 * expm1(log1p(cbrt(pow(cos(y), 2.0))))), cbrt(cos(y)), (z * 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 cbrt-unprod0.3

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

  8. Simplified0.3

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

  10. Applied expm1-log1p-u0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020066 +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))))