Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E

Average Accuracy: 99.6% → 99.7%

Time: 10.0s

Precision: binary64

Cost: 6848

Math

\[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]

↓

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

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* (- y x) 6.0) z)))

↓

(FPCore (x y z) :precision binary64 (fma 6.0 (* z (- y x)) x))

C

double code(double x, double y, double z) {
	return x + (((y - x) * 6.0) * z);
}

↓

double code(double x, double y, double z) {
	return fma(6.0, (z * (y - x)), x);
}

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(Float64(y - x) * 6.0) * z))
end

↓

function code(x, y, z)
	return fma(6.0, Float64(z * Float64(y - x)), x)
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * 6.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(6.0 * N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Target

Original	99.6%
Target	99.7%
Herbie	99.7%

\[x - \left(6 \cdot z\right) \cdot \left(x - y\right) \]

Derivation

1. Initial program 99.6%

   \[x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]

2. Simplified 99.7%

   \[\leadsto \color{blue}{x + \left(y - x\right) \cdot \left(6 \cdot z\right)} \]
   Proof

   [Start] 99.6	\[ x + \left(\left(y - x\right) \cdot 6\right) \cdot z \]
   associate-*l* [=>] 99.7	\[ x + \color{blue}{\left(y - x\right) \cdot \left(6 \cdot z\right)} \]

3. Taylor expanded in x around -inf 99.7%

   \[\leadsto \color{blue}{6 \cdot \left(y \cdot z\right) + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right)} \]

4. Simplified 99.7%

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

   [Start] 99.7	\[ 6 \cdot \left(y \cdot z\right) + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right) \]
   *-commutative [=>] 99.7	\[ \color{blue}{\left(y \cdot z\right) \cdot 6} + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right) \]
   associate-*r* [<=] 99.7	\[ \color{blue}{y \cdot \left(z \cdot 6\right)} + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right) \]
   *-commutative [<=] 99.7	\[ y \cdot \color{blue}{\left(6 \cdot z\right)} + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right) \]
   *-lft-identity [<=] 99.7	\[ \color{blue}{1 \cdot \left(y \cdot \left(6 \cdot z\right)\right)} + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right) \]
   *-commutative [<=] 99.7	\[ \color{blue}{\left(y \cdot \left(6 \cdot z\right)\right) \cdot 1} + -1 \cdot \left(\left(6 \cdot z - 1\right) \cdot x\right) \]
   mul-1-neg [=>] 99.7	\[ \left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 + \color{blue}{\left(-\left(6 \cdot z - 1\right) \cdot x\right)} \]
   unsub-neg [=>] 99.7	\[ \color{blue}{\left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - \left(6 \cdot z - 1\right) \cdot x} \]
   *-commutative [=>] 99.7	\[ \left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - \color{blue}{x \cdot \left(6 \cdot z - 1\right)} \]
   sub-neg [=>] 99.7	\[ \left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - x \cdot \color{blue}{\left(6 \cdot z + \left(-1\right)\right)} \]
   metadata-eval [=>] 99.7	\[ \left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - x \cdot \left(6 \cdot z + \color{blue}{-1}\right) \]
   distribute-rgt-in [=>] 99.7	\[ \left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - \color{blue}{\left(\left(6 \cdot z\right) \cdot x + -1 \cdot x\right)} \]
   neg-mul-1 [<=] 99.7	\[ \left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - \left(\left(6 \cdot z\right) \cdot x + \color{blue}{\left(-x\right)}\right) \]
   associate--r+ [=>] 99.7	\[ \color{blue}{\left(\left(y \cdot \left(6 \cdot z\right)\right) \cdot 1 - \left(6 \cdot z\right) \cdot x\right) - \left(-x\right)} \]

5. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	80.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-59} \lor \neg \left(z \leq 1.85 \cdot 10^{-6}\right):\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	80.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-62} \lor \neg \left(z \leq 0.0145\right):\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(x \cdot -6\right)\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00055 \lor \neg \left(z \leq 0.17\right):\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(6 \cdot y\right)\\ \end{array} \]

Alternative 4
Accuracy	98.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00055 \lor \neg \left(z \leq 0.17\right):\\ \;\;\;\;6 \cdot \left(z \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{0.16666666666666666}{y}}\\ \end{array} \]

Alternative 5
Accuracy	61.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00055 \lor \neg \left(z \leq 6200\right):\\ \;\;\;\;-6 \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	62.6%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-62} \lor \neg \left(z \leq 1.6 \cdot 10^{-6}\right):\\ \;\;\;\;6 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	62.6%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-61}:\\ \;\;\;\;z \cdot \left(6 \cdot y\right)\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;6 \cdot \left(z \cdot y\right)\\ \end{array} \]

Alternative 8
Accuracy	99.6%
Cost	576

\[x + z \cdot \left(6 \cdot \left(y - x\right)\right) \]

Alternative 9
Accuracy	99.7%
Cost	576

\[x + \left(y - x\right) \cdot \left(6 \cdot z\right) \]

Alternative 10
Accuracy	45.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSL:hsl from colour-2.3.3, E"
  :precision binary64

  :herbie-target
  (- x (* (* 6.0 z) (- x y)))

  (+ x (* (* (- y x) 6.0) z)))