?

Average Error: 0.4 → 0.2
Time: 15.2s
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 0.4

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

  2. Simplified0.2

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

    [Start]0.4

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

    +-commutative [=>]0.4

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

    associate-*l* [=>]0.2

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

    fma-def [=>]0.2

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

    sub-neg [=>]0.2

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

    distribute-lft-in [=>]0.2

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

    +-commutative [=>]0.2

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

    neg-mul-1 [=>]0.2

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

    associate-*r* [=>]0.2

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

    *-commutative [=>]0.2

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

    fma-def [=>]0.2

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

    metadata-eval [=>]0.2

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

    metadata-eval [=>]0.2

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

    metadata-eval [=>]0.2

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

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error0.3
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
Error32.7
Cost2169
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -410000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-57}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-209}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-251}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-252}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-167}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-30}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+51} \lor \neg \left(z \leq 2.9 \cdot 10^{+92}\right) \land z \leq 3.15 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
Alternative 3
Error32.8
Cost2168
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -410000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{-57}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-123}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-205}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-251}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-251}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-171}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 9.6 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-31}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+119}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
Alternative 4
Error32.8
Cost2168
\[\begin{array}{l} t_0 := y \cdot \left(z \cdot -6\right)\\ \mathbf{if}\;z \leq -410000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.6 \cdot 10^{-51}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-124}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-208}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-251}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-251}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.42 \cdot 10^{-170}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-35}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.62 \cdot 10^{+56}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
Alternative 5
Error32.8
Cost2168
\[\begin{array}{l} t_0 := z \cdot \left(y \cdot -6\right)\\ \mathbf{if}\;z \leq -410000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.9 \cdot 10^{-44}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-124}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-251}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-251}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-180}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 9 \cdot 10^{-35}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+47}:\\ \;\;\;\;-6 \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+118}:\\ \;\;\;\;x \cdot \left(z \cdot 6\right)\\ \mathbf{elif}\;z \leq 3.85 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(x \cdot z\right)\\ \end{array} \]
Alternative 6
Error21.5
Cost1900
\[\begin{array}{l} t_0 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.1 \cdot 10^{-47}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-123}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-205}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-249}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-251}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-168}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-35}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error32.4
Cost1772
\[\begin{array}{l} t_0 := -6 \cdot \left(y \cdot z\right)\\ \mathbf{if}\;z \leq -410000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-45}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-122}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-207}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.15 \cdot 10^{-250}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-252}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-174}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-29}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 0.66:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error20.9
Cost1768
\[\begin{array}{l} t_0 := x \cdot \left(-3 - z \cdot -6\right)\\ t_1 := -6 \cdot \left(\left(y - x\right) \cdot z\right)\\ \mathbf{if}\;z \leq -490000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{-123}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -3.6 \cdot 10^{-209}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-250}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-169}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{-61}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 17000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error20.9
Cost1768
\[\begin{array}{l} t_0 := x \cdot \left(-3 - z \cdot -6\right)\\ t_1 := z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{if}\;z \leq -1950000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.2 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-207}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq -2.35 \cdot 10^{-251}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-252}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-172}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-155}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-60}:\\ \;\;\;\;y \cdot 4\\ \mathbf{elif}\;z \leq 20500000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error0.3
Cost1088
\[x \cdot \left(1 + -6 \cdot \left(0.6666666666666666 - z\right)\right) + 6 \cdot \left(y \cdot \left(0.6666666666666666 - z\right)\right) \]
Alternative 11
Error17.4
Cost978
\[\begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-76} \lor \neg \left(x \leq 2.8 \cdot 10^{-127}\right) \land \left(x \leq 7 \cdot 10^{-92} \lor \neg \left(x \leq 4.3 \cdot 10^{-25}\right)\right):\\ \;\;\;\;x \cdot \left(-3 - z \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(4 + z \cdot -6\right)\\ \end{array} \]
Alternative 12
Error0.2
Cost960
\[\left(x + \left(y - x\right) \cdot \left(z \cdot -6\right)\right) + \left(y - x\right) \cdot 4 \]
Alternative 13
Error0.3
Cost832
\[x + \frac{\left(y - x\right) \cdot 6}{\frac{1}{0.6666666666666666 - z}} \]
Alternative 14
Error1.7
Cost713
\[\begin{array}{l} \mathbf{if}\;z \leq -0.56 \lor \neg \left(z \leq 0.58\right):\\ \;\;\;\;z \cdot \left(\left(y - x\right) \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 4 + x \cdot -3\\ \end{array} \]
Alternative 15
Error0.4
Cost704
\[x + \left(0.6666666666666666 - z\right) \cdot \left(\left(y - x\right) \cdot 6\right) \]
Alternative 16
Error33.3
Cost456
\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-68}:\\ \;\;\;\;x \cdot -3\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;y \cdot 4\\ \mathbf{else}:\\ \;\;\;\;x \cdot -3\\ \end{array} \]
Alternative 17
Error43.0
Cost192
\[y \cdot 4 \]

Error

Reproduce?

herbie shell --seed 2023023 
(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))))