?

Average Error: 0.69% → 0.3%
Time: 14.2s
Precision: binary64
Cost: 13248

?

\[x + \left(\left(y - x\right) \cdot 6\right) \cdot \left(\frac{2}{3} - z\right) \]
↓
\[\mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, 4\right), x\right) \]
(FPCore (x y z)
 :precision binary64
 (+ x (* (* (- y x) 6.0) (- (/ 2.0 3.0) z))))
↓
(FPCore (x y z) :precision binary64 (fma (- y x) (fma z -6.0 4.0) x))
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((y - x), fma(z, -6.0, 4.0), x);
}

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

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

Error?

Derivation?

  1. Initial program 0.69

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

  2. Simplified0.3

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

    [Start]0.69

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

    +-commutative [=>]0.69

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

    associate-*l* [=>]0.33

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

    fma-def [=>]0.32

    \[ \color{blue}{\mathsf{fma}\left(y - x, 6 \cdot \left(\frac{2}{3} - z\right), x\right)} \]

    sub-neg [=>]0.32

    \[ \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\left(\frac{2}{3} + \left(-z\right)\right)}, x\right) \]

    distribute-lft-in [=>]0.3

    \[ \mathsf{fma}\left(y - x, \color{blue}{6 \cdot \frac{2}{3} + 6 \cdot \left(-z\right)}, x\right) \]

    +-commutative [=>]0.3

    \[ \mathsf{fma}\left(y - x, \color{blue}{6 \cdot \left(-z\right) + 6 \cdot \frac{2}{3}}, x\right) \]

    neg-mul-1 [=>]0.3

    \[ \mathsf{fma}\left(y - x, 6 \cdot \color{blue}{\left(-1 \cdot z\right)} + 6 \cdot \frac{2}{3}, x\right) \]

    associate-*r* [=>]0.3

    \[ \mathsf{fma}\left(y - x, \color{blue}{\left(6 \cdot -1\right) \cdot z} + 6 \cdot \frac{2}{3}, x\right) \]

    *-commutative [=>]0.3

    \[ \mathsf{fma}\left(y - x, \color{blue}{z \cdot \left(6 \cdot -1\right)} + 6 \cdot \frac{2}{3}, x\right) \]

    fma-def [=>]0.3

    \[ \mathsf{fma}\left(y - x, \color{blue}{\mathsf{fma}\left(z, 6 \cdot -1, 6 \cdot \frac{2}{3}\right)}, x\right) \]

    metadata-eval [=>]0.3

    \[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, \color{blue}{-6}, 6 \cdot \frac{2}{3}\right), x\right) \]

    metadata-eval [=>]0.3

    \[ \mathsf{fma}\left(y - x, \mathsf{fma}\left(z, -6, 6 \cdot \color{blue}{0.6666666666666666}\right), x\right) \]

    metadata-eval [=>]0.3

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

  3. Final simplification0.3

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

Alternatives

Alternative 1
Error32.71%
Cost1504
\[\begin{array}{l} t_0 := x \cdot \left(-3 + z \cdot 6\right)\\ t_1 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -280000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-139}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-204}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-304}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10.5:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \mathbf{elif}\;z \leq 110000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error32.71%
Cost1504
\[\begin{array}{l} t_0 := x \cdot \left(-3 + z \cdot 6\right)\\ t_1 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{if}\;z \leq -255000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-138}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-203}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 12:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \mathbf{elif}\;z \leq 165000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error32.86%
Cost1504
\[\begin{array}{l} t_0 := x \cdot \left(-3 + z \cdot 6\right)\\ t_1 := y \cdot \left(4 + z \cdot -6\right)\\ t_2 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{if}\;z \leq -48000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.42 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{-143}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-204}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 9.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 150000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error33.02%
Cost1372
\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -0.058:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-55}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-138}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-202}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-304}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.15 \cdot 10^{-13}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2050000:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error51.28%
Cost1244
\[\begin{array}{l} t_0 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -0.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-55}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-137}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{-203}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-303}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+145}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
Alternative 6
Error51.52%
Cost1244
\[\begin{array}{l} t_0 := x \cdot \left(z \cdot 6\right)\\ \mathbf{if}\;z \leq -0.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-55}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-142}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.42 \cdot 10^{-202}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
Alternative 7
Error51.44%
Cost1244
\[\begin{array}{l} \mathbf{if}\;z \leq -0.5:\\ \;\;\;\;z \cdot \left(x \cdot 6\right)\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-55}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-137}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.26 \cdot 10^{-205}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+148}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \end{array} \]
Alternative 8
Error33.57%
Cost1240
\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -0.04:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-55}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-142}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{-202}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-305}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error50.92%
Cost1112
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -2450:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-55}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-139}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -2.05 \cdot 10^{-205}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-307}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error2.96%
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -0.6 \lor \neg \left(z \leq 0.63\right):\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \end{array} \]
Alternative 11
Error2.96%
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -0.55:\\ \;\;\;\;z \cdot \left(x \cdot 6 + y \cdot -6\right)\\ \mathbf{elif}\;z \leq 0.52:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]
Alternative 12
Error0.69%
Cost704
\[x + \left(0.6666666666666666 - z\right) \cdot \left(\left(y - x\right) \cdot 6\right) \]
Alternative 13
Error0.33%
Cost704
\[x + \left(y - x\right) \cdot \left(6 \cdot \left(0.6666666666666666 - z\right)\right) \]
Alternative 14
Error53.11%
Cost456
\[\begin{array}{l} \mathbf{if}\;y \leq -3700000000:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+87}:\\ \;\;\;\;x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4\\ \end{array} \]
Alternative 15
Error67.66%
Cost192
\[y \cdot 4 \]

Error

Reproduce?

herbie shell --seed 2023089 
(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))))