Average Error: 3.1 → 3.1
Time: 2.7s
Precision: binary64
\[x \cdot \left(1 - y \cdot z\right) \]
↓
\[x \cdot \mathsf{fma}\left(y, -z, 1\right) \]
(FPCore (x y z) :precision binary64 (* x (- 1.0 (* y z))))
↓
(FPCore (x y z) :precision binary64 (* x (fma y (- z) 1.0)))
double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}
↓
double code(double x, double y, double z) {
	return x * fma(y, -z, 1.0);
}

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(y * z)))
end
↓
function code(x, y, z)
	return Float64(x * fma(y, Float64(-z), 1.0))
end

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

x \cdot \left(1 - y \cdot z\right)
↓
x \cdot \mathsf{fma}\left(y, -z, 1\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 3.1

    \[x \cdot \left(1 - y \cdot z\right) \]

  2. Applied egg-rr4.3

    \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \left(1 - y \cdot z\right)}\right)}^{3}} \]

  3. Applied egg-rr3.4

    \[\leadsto \color{blue}{{\left(\sqrt[3]{1 - y \cdot z}\right)}^{2} \cdot \left(\sqrt[3]{1 - y \cdot z} \cdot x\right)} \]

  4. Taylor expanded in y around 0 4.4

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

  5. Simplified3.1

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

  6. Final simplification3.1

    \[\leadsto x \cdot \mathsf{fma}\left(y, -z, 1\right) \]

Reproduce

herbie shell --seed 2022162 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))