Average Error: 0.1 → 0.1
Time: 6.2s
Precision: binary64
\[x \cdot \cos y + z \cdot \sin y \]
↓
\[\mathsf{fma}\left(\sin y, z, \cos y \cdot x\right) \]
(FPCore (x y z) :precision binary64 (+ (* x (cos y)) (* z (sin y))))
↓
(FPCore (x y z) :precision binary64 (fma (sin y) z (* (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(sin(y), z, (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(sin(y), z, 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[(N[Sin[y], $MachinePrecision] * z + N[(N[Cos[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

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

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. Simplified0.1

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

  3. Taylor expanded in x around 0 0.1

    \[\leadsto \color{blue}{\cos y \cdot x + \sin y \cdot z} \]

  4. Simplified0.1

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

  5. Applied add-cube-cbrt_binary640.4

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

  6. Applied associate-*l*_binary640.4

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

  7. Applied add-cbrt-cube_binary640.4

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

  8. Simplified0.3

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

  9. Applied *-un-lft-identity_binary640.3

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

  10. Applied associate-*l*_binary640.3

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2022131 
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))