Average Error: 0.4 → 0.2
Time: 3.4s
Precision: binary64
\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(y, -6, x \cdot 6\right), z, x + \left(y - x\right) \cdot 4\right) \]
(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))
↓
(FPCore (x y z)
 :precision binary64
 (fma (fma y -6.0 (* x 6.0)) z (+ x (* (- y x) 4.0))))
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(fma(y, -6.0, (x * 6.0)), z, (x + ((y - x) * 4.0)));
}

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(fma(y, -6.0, Float64(x * 6.0)), z, Float64(x + Float64(Float64(y - x) * 4.0)))
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[(N[(y * -6.0 + N[(x * 6.0), $MachinePrecision]), $MachinePrecision] * z + N[(x + N[(N[(y - x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right)
↓
\mathsf{fma}\left(\mathsf{fma}\left(y, -6, x \cdot 6\right), z, x + \left(y - x\right) \cdot 4\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 z around 0 0.2

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

  4. Applied egg-rr0.2

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

  5. Applied egg-rr0.2

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

  6. Applied egg-rr0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022170 
(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))))