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

Average Error: 3.2 → 0.6

Time: 8.5s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \left(-z \cdot x\right)\\ \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+131}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- (* z x)))))
   (if (<= (* y z) -2e+131)
     t_0
     (if (<= (* y z) 5e+133) (* x (- 1.0 (* y z))) t_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 t_0 = y * -(z * x);
	double tmp;
	if ((y * z) <= -2e+131) {
		tmp = t_0;
	} else if ((y * z) <= 5e+133) {
		tmp = x * (1.0 - (y * z));
	} else {
		tmp = t_0;
	}
	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 = y * -(z * x)
    if ((y * z) <= (-2d+131)) then
        tmp = t_0
    else if ((y * z) <= 5d+133) then
        tmp = x * (1.0d0 - (y * z))
    else
        tmp = t_0
    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 = y * -(z * x);
	double tmp;
	if ((y * z) <= -2e+131) {
		tmp = t_0;
	} else if ((y * z) <= 5e+133) {
		tmp = x * (1.0 - (y * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = y * -(z * x)
	tmp = 0
	if (y * z) <= -2e+131:
		tmp = t_0
	elif (y * z) <= 5e+133:
		tmp = x * (1.0 - (y * z))
	else:
		tmp = t_0
	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(y * Float64(-Float64(z * x)))
	tmp = 0.0
	if (Float64(y * z) <= -2e+131)
		tmp = t_0;
	elseif (Float64(y * z) <= 5e+133)
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	else
		tmp = t_0;
	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 = y * -(z * x);
	tmp = 0.0;
	if ((y * z) <= -2e+131)
		tmp = t_0;
	elseif ((y * z) <= 5e+133)
		tmp = x * (1.0 - (y * z));
	else
		tmp = t_0;
	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[(y * (-N[(z * x), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -2e+131], t$95$0, If[LessEqual[N[(y * z), $MachinePrecision], 5e+133], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.9999999999999998e131 or 4.99999999999999961e133 < (*.f64 y z)

   1. Initial program 17.2

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

   2. Taylor expanded in y around inf 2.8

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

   3. Simplified 2.8

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

      [Start] 2.8	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      rational.json-simplify-43 [=>] 2.8	\[ \color{blue}{y \cdot \left(\left(z \cdot x\right) \cdot -1\right)} \]
      rational.json-simplify-9 [=>] 2.8	\[ y \cdot \color{blue}{\left(-z \cdot x\right)} \]

   if -1.9999999999999998e131 < (*.f64 y z) < 4.99999999999999961e133

   1. Initial program 0.1

      \[x \cdot \left(1 - y \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+131}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+133}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.2
Cost	912

\[\begin{array}{l} t_0 := x \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+146}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	17.1
Cost	912

\[\begin{array}{l} t_0 := y \cdot \left(-z \cdot x\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+146}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	16.9
Cost	912

\[\begin{array}{l} t_0 := z \cdot \left(-y \cdot x\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+61}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+147}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	24.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023068 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))