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

Average Error: 5.33% → 0.35%

Time: 6.1s

Precision: binary64

Cost: 1481

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(1 - y \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* y z)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+304)))
     (* z (* y (- x)))
     (- x (* x (* y z))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = x * (1.0 - (y * z));
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+304)) {
		tmp = z * (y * -x);
	} else {
		tmp = x - (x * (y * z));
	}
	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 t_0 = x * (1.0 - (y * z));
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e+304)) {
		tmp = z * (y * -x);
	} else {
		tmp = x - (x * (y * z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = x * (1.0 - (y * z))
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e+304):
		tmp = z * (y * -x)
	else:
		tmp = x - (x * (y * z))
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(y * z)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+304))
		tmp = Float64(z * Float64(y * Float64(-x)));
	else
		tmp = Float64(x - Float64(x * Float64(y * z)));
	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)
	t_0 = x * (1.0 - (y * z));
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e+304)))
		tmp = z * (y * -x);
	else
		tmp = x - (x * (y * z));
	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_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+304]], $MachinePrecision]], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], N[(x - N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 1 (*.f64 y z))) < -inf.0 or 1.9999999999999999e304 < (*.f64 x (-.f64 1 (*.f64 y z)))

   1. Initial program 95.18

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

   2. Taylor expanded in y around inf 4.47

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

   3. Simplified 3.77

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

      [Start] 4.47	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 4.47	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 98.36	\[ -\color{blue}{\left(y \cdot z\right) \cdot x} \]
      distribute-rgt-neg-in [=>] 98.36	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]
      *-commutative [=>] 98.36	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(-x\right) \]
      associate-*l* [=>] 3.77	\[ \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]

   if -inf.0 < (*.f64 x (-.f64 1 (*.f64 y z))) < 1.9999999999999999e304

   1. Initial program 0.16

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

   2. Taylor expanded in x around 0 0.16

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

   3. Simplified 8.48

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

      [Start] 0.16	\[ \left(1 - y \cdot z\right) \cdot x \]
      sub-neg [=>] 0.16	\[ \color{blue}{\left(1 + \left(-y \cdot z\right)\right)} \cdot x \]
      +-commutative [<=] 0.16	\[ \color{blue}{\left(\left(-y \cdot z\right) + 1\right)} \cdot x \]
      distribute-lft1-in [<=] 0.15	\[ \color{blue}{\left(-y \cdot z\right) \cdot x + x} \]
      distribute-lft-neg-out [=>] 0.15	\[ \color{blue}{\left(-\left(y \cdot z\right) \cdot x\right)} + x \]
      associate-*r* [<=] 8.48	\[ \left(-\color{blue}{y \cdot \left(z \cdot x\right)}\right) + x \]
      mul-1-neg [<=] 8.48	\[ \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot x\right)\right)} + x \]
      *-commutative [=>] 8.48	\[ \color{blue}{\left(y \cdot \left(z \cdot x\right)\right) \cdot -1} + x \]
      associate-*l* [=>] 8.48	\[ \color{blue}{y \cdot \left(\left(z \cdot x\right) \cdot -1\right)} + x \]
      fma-def [=>] 8.48	\[ \color{blue}{\mathsf{fma}\left(y, \left(z \cdot x\right) \cdot -1, x\right)} \]
      associate-*l* [=>] 8.48	\[ \mathsf{fma}\left(y, \color{blue}{z \cdot \left(x \cdot -1\right)}, x\right) \]
      *-commutative [=>] 8.48	\[ \mathsf{fma}\left(y, z \cdot \color{blue}{\left(-1 \cdot x\right)}, x\right) \]
      neg-mul-1 [<=] 8.48	\[ \mathsf{fma}\left(y, z \cdot \color{blue}{\left(-x\right)}, x\right) \]

   4. Taylor expanded in y around 0 8.48

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

   5. Simplified 0.15

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

      [Start] 8.48	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) + x \]
      associate-*r* [=>] 0.15	\[ -1 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} + x \]
      mul-1-neg [=>] 0.15	\[ \color{blue}{\left(-\left(y \cdot z\right) \cdot x\right)} + x \]
      +-commutative [<=] 0.15	\[ \color{blue}{x + \left(-\left(y \cdot z\right) \cdot x\right)} \]
      sub-neg [<=] 0.15	\[ \color{blue}{x - \left(y \cdot z\right) \cdot x} \]
      *-commutative [=>] 0.15	\[ x - \color{blue}{x \cdot \left(y \cdot z\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.35

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - y \cdot z\right) \leq -\infty \lor \neg \left(x \cdot \left(1 - y \cdot z\right) \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.36%
Cost	1481

\[\begin{array}{l} t_0 := x \cdot \left(1 - y \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{+304}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	37.02%
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+136} \lor \neg \left(y \leq -1.1 \cdot 10^{+81}\right) \land \left(y \leq -1.25 \cdot 10^{+30} \lor \neg \left(y \leq 5.2 \cdot 10^{-163}\right)\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	36.25%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+29} \lor \neg \left(y \leq 5.2 \cdot 10^{-163}\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	40.57%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))