Average Error: 3.4 → 1.7
Time: 2.6s
Precision: binary64
\[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
↓
\[\mathsf{fma}\left(z \cdot x, y + -1, x\right) \]
(FPCore (x y z) :precision binary64 (* x (- 1.0 (* (- 1.0 y) z))))
↓
(FPCore (x y z) :precision binary64 (fma (* z x) (+ y -1.0) x))
double code(double x, double y, double z) {
	return x * (1.0 - ((1.0 - y) * z));
}
↓
double code(double x, double y, double z) {
	return fma((z * x), (y + -1.0), x);
}

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

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

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original3.4
Target0.2
Herbie1.7
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

  1. Initial program 3.4

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

  2. Taylor expanded in x around 0 3.4

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

  3. Simplified1.7

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

  4. Final simplification1.7

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

Reproduce

herbie shell --seed 2022166 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))