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

Average Error: 3.2 → 0.4

Time: 7.5s

Precision: binary64

Cost: 1612

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 1.005:\\ \;\;\;\;\frac{x}{\frac{1}{t_0}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+194}:\\ \;\;\;\;t_0 \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- 1.0 (* y z))))
   (if (<= t_0 -5e+169)
     (* y (* (- z) x))
     (if (<= t_0 1.005)
       (/ x (/ 1.0 t_0))
       (if (<= t_0 5e+194) (* t_0 x) (* z (* y (- 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 t_0 = 1.0 - (y * z);
	double tmp;
	if (t_0 <= -5e+169) {
		tmp = y * (-z * x);
	} else if (t_0 <= 1.005) {
		tmp = x / (1.0 / t_0);
	} else if (t_0 <= 5e+194) {
		tmp = t_0 * x;
	} else {
		tmp = z * (y * -x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (1.0d0 - (y * z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y * z)
    if (t_0 <= (-5d+169)) then
        tmp = y * (-z * x)
    else if (t_0 <= 1.005d0) then
        tmp = x / (1.0d0 / t_0)
    else if (t_0 <= 5d+194) then
        tmp = t_0 * x
    else
        tmp = z * (y * -x)
    end if
    code = tmp
end function

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 = 1.0 - (y * z);
	double tmp;
	if (t_0 <= -5e+169) {
		tmp = y * (-z * x);
	} else if (t_0 <= 1.005) {
		tmp = x / (1.0 / t_0);
	} else if (t_0 <= 5e+194) {
		tmp = t_0 * x;
	} else {
		tmp = z * (y * -x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = 1.0 - (y * z)
	tmp = 0
	if t_0 <= -5e+169:
		tmp = y * (-z * x)
	elif t_0 <= 1.005:
		tmp = x / (1.0 / t_0)
	elif t_0 <= 5e+194:
		tmp = t_0 * x
	else:
		tmp = z * (y * -x)
	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(1.0 - Float64(y * z))
	tmp = 0.0
	if (t_0 <= -5e+169)
		tmp = Float64(y * Float64(Float64(-z) * x));
	elseif (t_0 <= 1.005)
		tmp = Float64(x / Float64(1.0 / t_0));
	elseif (t_0 <= 5e+194)
		tmp = Float64(t_0 * x);
	else
		tmp = Float64(z * Float64(y * Float64(-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)
	t_0 = 1.0 - (y * z);
	tmp = 0.0;
	if (t_0 <= -5e+169)
		tmp = y * (-z * x);
	elseif (t_0 <= 1.005)
		tmp = x / (1.0 / t_0);
	elseif (t_0 <= 5e+194)
		tmp = t_0 * x;
	else
		tmp = z * (y * -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_] := Block[{t$95$0 = N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+169], N[(y * N[((-z) * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.005], N[(x / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+194], N[(t$95$0 * x), $MachinePrecision], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 1 (*.f64 y z)) < -5.00000000000000017e169

   1. Initial program 20.5

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

   2. Taylor expanded in y around inf 2.1

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

   3. Simplified 2.1

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

      [Start] 2.1	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 2.1	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 2.1	\[ \color{blue}{y \cdot \left(-z \cdot x\right)} \]
      distribute-lft-neg-in [=>] 2.1	\[ y \cdot \color{blue}{\left(\left(-z\right) \cdot x\right)} \]

   if -5.00000000000000017e169 < (-.f64 1 (*.f64 y z)) < 1.0049999999999999

   1. Initial program 0.1

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

   2. Applied egg-rr 40.7

      \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot \left(1 - y \cdot z\right)\right)}^{3}}} \]

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 - y \cdot z}}} \]

   if 1.0049999999999999 < (-.f64 1 (*.f64 y z)) < 4.99999999999999989e194

   1. Initial program 0.3

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

   if 4.99999999999999989e194 < (-.f64 1 (*.f64 y z))

   1. Initial program 26.1

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

   2. Taylor expanded in y around inf 2.1

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

   3. Simplified 2.1

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

      [Start] 2.1	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 2.1	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 26.1	\[ -\color{blue}{\left(y \cdot z\right) \cdot x} \]
      distribute-rgt-neg-in [=>] 26.1	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]
      *-commutative [=>] 26.1	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(-x\right) \]
      associate-*l* [=>] 2.1	\[ \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - y \cdot z \leq -5 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;1 - y \cdot z \leq 1.005:\\ \;\;\;\;\frac{x}{\frac{1}{1 - y \cdot z}}\\ \mathbf{elif}\;1 - y \cdot z \leq 5 \cdot 10^{+194}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.2
Cost	1179

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+125} \lor \neg \left(y \leq -1.75 \cdot 10^{+64} \lor \neg \left(y \leq -5.4 \cdot 10^{+43}\right) \land \left(y \leq -80000 \lor \neg \left(y \leq -2.05 \cdot 10^{-13}\right) \land y \leq 2.55 \cdot 10^{-144}\right)\right):\\ \;\;\;\;x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	1.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+198}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y \cdot z\right) \cdot x\\ \end{array} \]

Alternative 3
Error	16.9
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-127} \lor \neg \left(z \leq 2.2 \cdot 10^{+108}\right):\\ \;\;\;\;y \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	25.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023054 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))