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

Average Error: 3.3 → 0.7

Time: 4.4s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+296}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(-y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -1e+296)
   (* z (* y (- x)))
   (if (<= (* y z) 5e+94) (* x (- 1.0 (* y z))) (* (* z x) (- y)))))

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+296) {
		tmp = z * (y * -x);
	} else if ((y * z) <= 5e+94) {
		tmp = x * (1.0 - (y * z));
	} else {
		tmp = (z * x) * -y;
	}
	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) :: tmp
    if ((y * z) <= (-1d+296)) then
        tmp = z * (y * -x)
    else if ((y * z) <= 5d+94) then
        tmp = x * (1.0d0 - (y * z))
    else
        tmp = (z * x) * -y
    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 tmp;
	if ((y * z) <= -1e+296) {
		tmp = z * (y * -x);
	} else if ((y * z) <= 5e+94) {
		tmp = x * (1.0 - (y * z));
	} else {
		tmp = (z * x) * -y;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (y * z) <= -1e+296:
		tmp = z * (y * -x)
	elif (y * z) <= 5e+94:
		tmp = x * (1.0 - (y * z))
	else:
		tmp = (z * x) * -y
	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+296)
		tmp = Float64(z * Float64(y * Float64(-x)));
	elseif (Float64(y * z) <= 5e+94)
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	else
		tmp = Float64(Float64(z * x) * Float64(-y));
	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) <= -1e+296)
		tmp = z * (y * -x);
	elseif ((y * z) <= 5e+94)
		tmp = x * (1.0 - (y * z));
	else
		tmp = (z * x) * -y;
	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[LessEqual[N[(y * z), $MachinePrecision], -1e+296], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e+94], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * x), $MachinePrecision] * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -9.99999999999999981e295

   1. Initial program 57.1

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

   2. Taylor expanded in y around inf 0.2

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

   3. Simplified 0.2

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

      [Start] 0.2	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 0.2	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 0.2	\[ -\color{blue}{\left(z \cdot x\right) \cdot y} \]
      associate-*l* [=>] 0.2	\[ -\color{blue}{z \cdot \left(x \cdot y\right)} \]
      distribute-rgt-neg-in [=>] 0.2	\[ \color{blue}{z \cdot \left(-x \cdot y\right)} \]
      *-commutative [<=] 0.2	\[ z \cdot \left(-\color{blue}{y \cdot x}\right) \]
      distribute-rgt-neg-in [=>] 0.2	\[ z \cdot \color{blue}{\left(y \cdot \left(-x\right)\right)} \]

   if -9.99999999999999981e295 < (*.f64 y z) < 5.0000000000000001e94

   1. Initial program 0.1

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

   if 5.0000000000000001e94 < (*.f64 y z)

   1. Initial program 14.4

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

   2. Simplified 14.4

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

      [Start] 14.4	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 14.4	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 14.4	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 14.4	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 14.4	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

   3. Taylor expanded in z around inf 5.2

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

   4. Simplified 5.2

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

      [Start] 5.2	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      associate-*r* [=>] 5.2	\[ \color{blue}{\left(-1 \cdot y\right) \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 5.2	\[ \color{blue}{\left(z \cdot x\right) \cdot \left(-1 \cdot y\right)} \]
      mul-1-neg [=>] 5.2	\[ \left(z \cdot x\right) \cdot \color{blue}{\left(-y\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+296}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(-y\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	19.6
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+98} \lor \neg \left(y \leq -6.5 \cdot 10^{+65}\right) \land \left(y \leq -0.11 \lor \neg \left(y \leq 4.6 \cdot 10^{-103}\right)\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	18.5
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+78} \lor \neg \left(y \leq -6.2 \cdot 10^{+64}\right) \land \left(y \leq -200 \lor \neg \left(y \leq 5.2 \cdot 10^{-23}\right)\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	18.1
Cost	912

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -850:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \left(-y\right)\\ \end{array} \]

Alternative 4
Error	25.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))