Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I

Average Error: 3.4 → 0.6

Time: 7.2s

Precision: binary64

Cost: 7305

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+220} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+107}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, -y, 1\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (- 1.0 (* y z))))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) -1e+220) (not (<= (* y z) 2e+107)))
   (* y (* x (- z)))
   (* x (fma z (- y) 1.0))))

C

double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}

↓

double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -1e+220) || !((y * z) <= 2e+107)) {
		tmp = y * (x * -z);
	} else {
		tmp = x * fma(z, -y, 1.0);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if ((Float64(y * z) <= -1e+220) || !(Float64(y * z) <= 2e+107))
		tmp = Float64(y * Float64(x * Float64(-z)));
	else
		tmp = Float64(x * fma(z, Float64(-y), 1.0));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -1e+220], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e+107]], $MachinePrecision]], N[(y * N[(x * (-z)), $MachinePrecision]), $MachinePrecision], N[(x * N[(z * (-y) + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1e220 or 1.9999999999999999e107 < (*.f64 y z)

   1. Initial program 19.7

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

   2. Simplified 19.7

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

      [Start] 19.7	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 19.7	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 19.7	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 19.7	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 19.7	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

   3. Taylor expanded in z around inf 3.1

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

   4. Simplified 3.1

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

      [Start] 3.1	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      associate-*r* [=>] 3.1	\[ \color{blue}{\left(-1 \cdot y\right) \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 3.1	\[ \color{blue}{\left(z \cdot x\right) \cdot \left(-1 \cdot y\right)} \]
      mul-1-neg [=>] 3.1	\[ \left(z \cdot x\right) \cdot \color{blue}{\left(-y\right)} \]

   if -1e220 < (*.f64 y z) < 1.9999999999999999e107

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 0.1	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 0.1	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 0.1	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 0.1	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+220} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+107}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, -y, 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.7
Cost	1176

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-34}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+220} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+107}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]

Alternative 3
Error	17.7
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-124} \lor \neg \left(z \leq 9 \cdot 10^{+89}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	16.4
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-124}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 10^{+120}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 5
Error	24.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023073 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))