?

Average Error: 0.62% → 0.31%
Time: 15.3s
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 0.62

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

  2. Simplified0.31

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

    [Start]0.62

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

    +-commutative [=>]0.62

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

    associate-*l* [=>]0.35

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

    fma-def [=>]0.33

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

    sub-neg [=>]0.33

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

    distribute-lft-in [=>]0.32

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

    +-commutative [=>]0.32

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

    neg-mul-1 [=>]0.32

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

    associate-*r* [=>]0.32

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

    *-commutative [=>]0.32

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

    fma-def [=>]0.31

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

    metadata-eval [=>]0.31

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

    metadata-eval [=>]0.31

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

    metadata-eval [=>]0.31

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

  3. Final simplification0.31

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

Alternatives

Alternative 1
Error0.47%
Cost7360
\[\mathsf{fma}\left(1 + -6 \cdot \left(0.6666666666666666 - z\right), x, \left(0.6666666666666666 - z\right) \cdot \left(y \cdot 6\right)\right) \]
Alternative 2
Error0.63%
Cost6976
\[\mathsf{fma}\left(\frac{y - x}{0.16666666666666666}, 0.6666666666666666 - z, x\right) \]
Alternative 3
Error51.36%
Cost1376
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ t_1 := 6 \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -27500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.75 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-293}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-239}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error51.43%
Cost1376
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ t_1 := x \cdot \left(z \cdot 6\right)\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -27500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.16 \cdot 10^{-137}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.22 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-293}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-241}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error51.28%
Cost1376
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{+217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -460000:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-138}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-292}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.45 \cdot 10^{-239}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot 6\right)\\ \end{array} \]
Alternative 6
Error51.38%
Cost1376
\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+216}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -27500:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-135}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-292}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-240}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+61}:\\ \;\;\;\;y \cdot \left(z \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(x \cdot 6\right)\\ \end{array} \]
Alternative 7
Error50.98%
Cost1244
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -0.052:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-89}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-132}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-292}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-240}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-12}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error28.53%
Cost1243
\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+77} \lor \neg \left(x \leq -2.2 \cdot 10^{+21}\right) \land \left(x \leq -1.65 \cdot 10^{-23} \lor \neg \left(x \leq 5.4 \cdot 10^{-89}\right) \land \left(x \leq 7.5 \cdot 10^{-45} \lor \neg \left(x \leq 3.3 \cdot 10^{+17}\right)\right)\right):\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ \end{array} \]
Alternative 9
Error28.44%
Cost1243
\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+77} \lor \neg \left(x \leq -2.6 \cdot 10^{+22}\right) \land \left(x \leq -2.4 \cdot 10^{-24} \lor \neg \left(x \leq 5.2 \cdot 10^{-89}\right) \land \left(x \leq 2.4 \cdot 10^{-42} \lor \neg \left(x \leq 6.2 \cdot 10^{+15}\right)\right)\right):\\ \;\;\;\;x \cdot \left(-3 + z \cdot 6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(4 + z \cdot -6\right)\\ \end{array} \]
Alternative 10
Error33.08%
Cost1240
\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-133}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -5.4 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.46 \cdot 10^{-291}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-239}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.6:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error32.22%
Cost1240
\[\begin{array}{l} t_0 := 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right)\\ t_1 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -1450000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-163}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-292}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-238}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 12000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 12
Error0.48%
Cost832
\[x + \frac{\left(y - x\right) \cdot 6}{\frac{1}{0.6666666666666666 - z}} \]
Alternative 13
Error2.69%
Cost712
\[\begin{array}{l} \mathbf{if}\;z \leq -0.58:\\ \;\;\;\;-6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{elif}\;z \leq 0.65:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \end{array} \]
Alternative 14
Error0.62%
Cost704
\[x + \left(0.6666666666666666 - z\right) \cdot \left(\left(y - x\right) \cdot 6\right) \]
Alternative 15
Error53.68%
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq 2750:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot -3\\ \end{array} \]
Alternative 16
Error67.97%
Cost192
\[y \cdot 4 \]

Error

Reproduce?

herbie shell --seed 2023088 
(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))))