Average Error: 0.1 → 0.1
Time: 7.9s
Precision: binary64
\[x \cdot \cos y - z \cdot \sin y \]
↓
\[\mathsf{fma}\left(x, \cos y, \mathsf{fma}\left(z, -\sin y, 0\right)\right) \]
x \cdot \cos y - z \cdot \sin y
↓
\mathsf{fma}\left(x, \cos y, \mathsf{fma}\left(z, -\sin y, 0\right)\right)

(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z) :precision binary64 (fma x (cos y) (fma z (- (sin y)) 0.0)))
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, cos(y), fma(z, -sin(y), 0.0));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x \cdot \cos y - z \cdot \sin y \]

  2. Applied egg-rr0.2

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

  3. Taylor expanded in z around 0 0.1

    \[\leadsto \mathsf{fma}\left(x, \cos y, \mathsf{fma}\left(z, -\sin y, \color{blue}{0}\right)\right) \]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \cos y, \mathsf{fma}\left(z, -\sin y, 0\right)\right) \]

Reproduce

herbie shell --seed 2022125 
(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))))