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

Average Error: 3.4 → 2.3

Time: 8.8s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -1e+98) (* z (* y (- 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) <= -1e+98) {
		tmp = z * (y * -x);
	} else {
		tmp = x - ((y * z) * 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) :: tmp
    if ((y * z) <= (-1d+98)) then
        tmp = z * (y * -x)
    else
        tmp = x - ((y * z) * 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 tmp;
	if ((y * z) <= -1e+98) {
		tmp = z * (y * -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) <= -1e+98:
		tmp = z * (y * -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) <= -1e+98)
		tmp = Float64(z * Float64(y * 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) <= -1e+98)
		tmp = z * (y * -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[LessEqual[N[(y * z), $MachinePrecision], -1e+98], N[(z * N[(y * (-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) < -9.99999999999999998e97

   1. Initial program 13.7

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

   2. Taylor expanded in y around inf 3.4

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

   3. Simplified 4.6

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

      [Start] 3.4	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 3.4	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 13.7	\[ -\color{blue}{\left(y \cdot z\right) \cdot x} \]
      distribute-rgt-neg-in [=>] 13.7	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]
      *-commutative [=>] 13.7	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(-x\right) \]
      associate-*l* [=>] 4.6	\[ \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]

   if -9.99999999999999998e97 < (*.f64 y z)

   1. Initial program 2.1

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

   2. Taylor expanded in x around 0 2.1

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

   3. Simplified 5.2

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

      [Start] 2.1	\[ \left(1 - y \cdot z\right) \cdot x \]
      *-commutative [<=] 2.1	\[ \color{blue}{x \cdot \left(1 - y \cdot z\right)} \]
      distribute-rgt-out-- [<=] 2.0	\[ \color{blue}{1 \cdot x - \left(y \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 2.0	\[ \color{blue}{x} - \left(y \cdot z\right) \cdot x \]
      associate-*r* [<=] 5.2	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]

   4. Applied egg-rr 20.7

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

   5. Simplified 2.0

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

      [Start] 20.7	\[ x - \left(e^{\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)} - 1\right) \]
      expm1-def [=>] 15.1	\[ x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)\right)} \]
      expm1-log1p [=>] 5.2	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 2.0	\[ x - \color{blue}{\left(y \cdot z\right) \cdot x} \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+98}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]

Alternative 2
Error	17.8
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+80} \lor \neg \left(y \leq 2.2 \cdot 10^{-82}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	17.7
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-49}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+83}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \end{array} \]

Alternative 4
Error	16.8
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+79}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \end{array} \]

Alternative 5
Error	25.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))