Average Error: 0.1 → 0.1
Time: 4.8s
Precision: binary64
\[x \cdot \cos y - z \cdot \sin y \]
↓
\[\mathsf{fma}\left(z, -\sin y, \cos y \cdot x\right) \]
(FPCore (x y z) :precision binary64 (- (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z) :precision binary64 (fma z (- (sin y)) (* (cos y) x)))
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(z, -sin(y), (cos(y) * x));
}

function code(x, y, z)
	return Float64(Float64(x * cos(y)) - Float64(z * sin(y)))
end
↓
function code(x, y, z)
	return fma(z, Float64(-sin(y)), Float64(cos(y) * x))
end

code[x_, y_, z_] := N[(N[(x * N[Cos[y], $MachinePrecision]), $MachinePrecision] - N[(z * N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(z * (-N[Sin[y], $MachinePrecision]) + N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

x \cdot \cos y - z \cdot \sin y
↓
\mathsf{fma}\left(z, -\sin y, \cos y \cdot x\right)

Error

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded in x around 0 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2022185 
(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))))