Average Error: 0.4 → 0.2
Time: 3.5s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
↓
\[\mathsf{fma}\left(4, y, z \cdot \left(x \cdot 6\right) - \mathsf{fma}\left(x, 3, z \cdot \left(y \cdot 6\right)\right)\right) \]
(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))
↓
(FPCore (x y z)
 :precision binary64
 (fma 4.0 y (- (* z (* x 6.0)) (fma x 3.0 (* z (* y 6.0))))))
double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * ((2.0 / 3.0) - z));
}
↓
double code(double x, double y, double z) {
	return fma(4.0, y, ((z * (x * 6.0)) - fma(x, 3.0, (z * (y * 6.0)))));
}

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * Float64(Float64(2.0 / 3.0) - z)))
end
↓
function code(x, y, z)
	return fma(4.0, y, Float64(Float64(z * Float64(x * 6.0)) - fma(x, 3.0, Float64(z * Float64(y * 6.0)))))
end

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * N[(N[(2.0 / 3.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := N[(4.0 * y + N[(N[(z * N[(x * 6.0), $MachinePrecision]), $MachinePrecision] - N[(x * 3.0 + N[(z * N[(y * 6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(4, y, z \cdot \left(x \cdot 6\right) - \mathsf{fma}\left(x, 3, z \cdot \left(y \cdot 6\right)\right)\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.4

    \[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, 4\right), x\right)} \]

  3. Taylor expanded in y around 0 0.2

    \[\leadsto \color{blue}{\left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) - \left(6 \cdot \left(y \cdot z\right) + 3 \cdot x\right)} \]

  4. Applied egg-rr0.2

    \[\leadsto \left(4 \cdot y + 6 \cdot \left(z \cdot x\right)\right) - \color{blue}{\mathsf{fma}\left(x, 3, z \cdot \left(y \cdot 6\right)\right)} \]

  5. Applied egg-rr0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, y, z \cdot \left(x \cdot 6\right) - \mathsf{fma}\left(x, 3, z \cdot \left(y \cdot 6\right)\right)\right)} \]

  6. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(4, y, z \cdot \left(x \cdot 6\right) - \mathsf{fma}\left(x, 3, z \cdot \left(y \cdot 6\right)\right)\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, D"
  :precision binary64
  (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))