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

Average Error: 3.3 → 0.3

Time: 5.2s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+285}:\\ \;\;\;\;x - \frac{y}{\frac{\frac{1}{x}}{z}}\\ \mathbf{elif}\;y \cdot z \leq 10^{+165}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -2e+285)
   (- x (/ y (/ (/ 1.0 x) z)))
   (if (<= (* y z) 1e+165) (* x (- 1.0 (* y z))) (* 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 tmp;
	if ((y * z) <= -2e+285) {
		tmp = x - (y / ((1.0 / x) / z));
	} else if ((y * z) <= 1e+165) {
		tmp = x * (1.0 - (y * z));
	} 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) :: tmp
    if ((y * z) <= (-2d+285)) then
        tmp = x - (y / ((1.0d0 / x) / z))
    else if ((y * z) <= 1d+165) then
        tmp = x * (1.0d0 - (y * z))
    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 tmp;
	if ((y * z) <= -2e+285) {
		tmp = x - (y / ((1.0 / x) / z));
	} else if ((y * z) <= 1e+165) {
		tmp = x * (1.0 - (y * z));
	} else {
		tmp = z * (y * -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) <= -2e+285:
		tmp = x - (y / ((1.0 / x) / z))
	elif (y * z) <= 1e+165:
		tmp = x * (1.0 - (y * z))
	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)
	tmp = 0.0
	if (Float64(y * z) <= -2e+285)
		tmp = Float64(x - Float64(y / Float64(Float64(1.0 / x) / z)));
	elseif (Float64(y * z) <= 1e+165)
		tmp = Float64(x * Float64(1.0 - Float64(y * z)));
	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)
	tmp = 0.0;
	if ((y * z) <= -2e+285)
		tmp = x - (y / ((1.0 / x) / z));
	elseif ((y * z) <= 1e+165)
		tmp = x * (1.0 - (y * z));
	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_] := If[LessEqual[N[(y * z), $MachinePrecision], -2e+285], N[(x - N[(y / N[(N[(1.0 / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 1e+165], N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -2e285

   1. Initial program 48.4

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

   2. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\frac{\left(1 - {\left(y \cdot z\right)}^{2}\right) \cdot x}{1 + y \cdot z}} \]

   3. Applied egg-rr 48.4

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

   4. Applied egg-rr 0.4

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

   if -2e285 < (*.f64 y z) < 9.99999999999999899e164

   1. Initial program 0.1

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

   if 9.99999999999999899e164 < (*.f64 y z)

   1. Initial program 20.4

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

   2. Taylor expanded in y around inf 2.2

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

   3. Simplified 2.1

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

      [Start] 2.2	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 2.2	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 20.4	\[ -\color{blue}{\left(y \cdot z\right) \cdot x} \]
      distribute-rgt-neg-in [=>] 20.4	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]
      *-commutative [=>] 20.4	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(-x\right) \]
      associate-*l* [=>] 2.1	\[ \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+285}:\\ \;\;\;\;x - \frac{y}{\frac{\frac{1}{x}}{z}}\\ \mathbf{elif}\;y \cdot z \leq 10^{+165}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -\infty \lor \neg \left(y \cdot z \leq 10^{+165}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \end{array} \]

Alternative 2
Error	19.0
Cost	914

\[\begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-143} \lor \neg \left(z \leq 1.65 \cdot 10^{+127} \lor \neg \left(z \leq 3.5 \cdot 10^{+169}\right) \land z \leq 5.6 \cdot 10^{+187}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	18.0
Cost	912

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+169}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+184}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	17.6
Cost	912

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-103}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+127}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+169}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 5
Error	26.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023059 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))