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

Average Error: 3.3 → 0.2

Time: 7.1s

Precision: binary64

Cost: 969

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -\infty \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+177}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* y z) (- INFINITY)) (not (<= (* y z) 2e+177)))
   (* y (* z (- x)))
   (- x (* (* y z) x))))

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) <= -((double) INFINITY)) || !((y * z) <= 2e+177)) {
		tmp = y * (z * -x);
	} else {
		tmp = x - ((y * z) * x);
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (((y * z) <= -Double.POSITIVE_INFINITY) || !((y * z) <= 2e+177)) {
		tmp = y * (z * -x);
	} else {
		tmp = x - ((y * z) * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * (1.0 - (y * z))

↓

def code(x, y, z):
	tmp = 0
	if ((y * z) <= -math.inf) or not ((y * z) <= 2e+177):
		tmp = y * (z * -x)
	else:
		tmp = x - ((y * z) * x)
	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) <= Float64(-Inf)) || !(Float64(y * z) <= 2e+177))
		tmp = Float64(y * Float64(z * Float64(-x)));
	else
		tmp = Float64(x - Float64(Float64(y * z) * x));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - (y * z));
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((y * z) <= -Inf) || ~(((y * z) <= 2e+177)))
		tmp = y * (z * -x);
	else
		tmp = x - ((y * z) * x);
	end
	tmp_2 = 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], (-Infinity)], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e+177]], $MachinePrecision]], N[(y * N[(z * (-x)), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -inf.0 or 2e177 < (*.f64 y z)

   1. Initial program 33.7

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

   2. Taylor expanded in y around inf 1.7

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

   3. Simplified 1.7

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

      [Start] 1.7	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 1.7	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 1.7	\[ \color{blue}{y \cdot \left(-z \cdot x\right)} \]
      distribute-lft-neg-in [=>] 1.7	\[ y \cdot \color{blue}{\left(\left(-z\right) \cdot x\right)} \]

   if -inf.0 < (*.f64 y z) < 2e177

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 0.1

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

   3. Simplified 5.0

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

      [Start] 0.1	\[ \left(1 - y \cdot z\right) \cdot x \]
      *-commutative [<=] 0.1	\[ \color{blue}{x \cdot \left(1 - y \cdot z\right)} \]
      distribute-rgt-out-- [<=] 0.1	\[ \color{blue}{1 \cdot x - \left(y \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 0.1	\[ \color{blue}{x} - \left(y \cdot z\right) \cdot x \]
      associate-*r* [<=] 5.0	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]

   4. Applied egg-rr 20.9

      \[\leadsto x - \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)} - 1\right)} \]

   5. Simplified 0.1

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

      [Start] 20.9	\[ x - \left(e^{\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)} - 1\right) \]
      expm1-def [=>] 14.6	\[ x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)\right)} \]
      expm1-log1p [=>] 5.0	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 0.1	\[ x - \color{blue}{\left(y \cdot z\right) \cdot x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -\infty \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+177}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -\infty \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+177}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]

Alternative 2
Error	18.1
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-29} \lor \neg \left(z \leq 4 \cdot 10^{+94}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	16.6
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{-30} \lor \neg \left(z \leq 2.5 \cdot 10^{+94}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	24.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))