Average Error: 0.1 → 0.1
Time: 4.9s
Precision: binary64
\[x \cdot \sin y + z \cdot \cos y \]
↓
\[\mathsf{fma}\left(x, \sin y, z \cdot \left({1}^{0.3333333333333333} \cdot \cos y\right)\right) \]
(FPCore (x y z) :precision binary64 (+ (* x (sin y)) (* z (cos y))))
↓
(FPCore (x y z)
 :precision binary64
 (fma x (sin y) (* z (* (pow 1.0 0.3333333333333333) (cos y)))))
double code(double x, double y, double z) {
	return (x * sin(y)) + (z * cos(y));
}
↓
double code(double x, double y, double z) {
	return fma(x, sin(y), (z * (pow(1.0, 0.3333333333333333) * cos(y))));
}

function code(x, y, z)
	return Float64(Float64(x * sin(y)) + Float64(z * cos(y)))
end
↓
function code(x, y, z)
	return fma(x, sin(y), Float64(z * Float64((1.0 ^ 0.3333333333333333) * cos(y))))
end

code[x_, y_, z_] := N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(x * N[Sin[y], $MachinePrecision] + N[(z * N[(N[Power[1.0, 0.3333333333333333], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(x, \sin y, z \cdot \left({1}^{0.3333333333333333} \cdot \cos y\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Applied egg-rr0.2

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

  4. Applied egg-rr0.1

    \[\leadsto \mathsf{fma}\left(x, \sin y, z \cdot \color{blue}{\left({1}^{0.3333333333333333} \cdot \cos y\right)}\right) \]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \sin y, z \cdot \left({1}^{0.3333333333333333} \cdot \cos y\right)\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  :precision binary64
  (+ (* x (sin y)) (* z (cos y))))