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

Percentage Accurate: 96.2% → 99.7%

Time: 7.1s

Precision: binary64

Cost: 968

• Unsound rule application detected in e-graph. Results may not be sound.

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+239}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -5e+239)
   (* y (- (* z x)))
   (if (<= (* y z) 5e+168) (* 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) <= -5e+239) {
		tmp = y * -(z * x);
	} else if ((y * z) <= 5e+168) {
		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) <= (-5d+239)) then
        tmp = y * -(z * x)
    else if ((y * z) <= 5d+168) 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) <= -5e+239) {
		tmp = y * -(z * x);
	} else if ((y * z) <= 5e+168) {
		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) <= -5e+239:
		tmp = y * -(z * x)
	elif (y * z) <= 5e+168:
		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) <= -5e+239)
		tmp = Float64(y * Float64(-Float64(z * x)));
	elseif (Float64(y * z) <= 5e+168)
		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) <= -5e+239)
		tmp = y * -(z * x);
	elseif ((y * z) <= 5e+168)
		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], -5e+239], N[(y * (-N[(z * x), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e+168], 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) < -5.00000000000000007e239

   1. Initial program 77.1%

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

   2. Taylor expanded in y around inf 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{y \cdot \left(x \cdot \left(-z\right)\right)} \]
      Step-by-step derivation

      [Start] 99.9%	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 99.9%	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      distribute-rgt-neg-in [=>] 99.9%	\[ \color{blue}{y \cdot \left(-z \cdot x\right)} \]
      distribute-lft-neg-out [<=] 99.9%	\[ y \cdot \color{blue}{\left(\left(-z\right) \cdot x\right)} \]
      *-commutative [=>] 99.9%	\[ y \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)} \]

   if -5.00000000000000007e239 < (*.f64 y z) < 4.99999999999999967e168

   1. Initial program 99.9%

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

   if 4.99999999999999967e168 < (*.f64 y z)

   1. Initial program 86.2%

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

   2. Taylor expanded in x around 0 86.2%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x - y \cdot \left(z \cdot x\right)} \]
      Step-by-step derivation

      [Start] 86.2%	\[ \left(1 - y \cdot z\right) \cdot x \]
      *-commutative [=>] 86.2%	\[ \color{blue}{x \cdot \left(1 - y \cdot z\right)} \]
      distribute-rgt-out-- [<=] 86.2%	\[ \color{blue}{1 \cdot x - \left(y \cdot z\right) \cdot x} \]
      associate-*r* [<=] 99.8%	\[ 1 \cdot x - \color{blue}{y \cdot \left(z \cdot x\right)} \]
      *-lft-identity [=>] 99.8%	\[ \color{blue}{x} - y \cdot \left(z \cdot x\right) \]

   4. Applied egg-rr 20.2%

      \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\left(-y\right) \cdot \left(z \cdot x\right)\right)}^{3}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)}} \]
      Step-by-step derivation

      [Start] 99.8%	\[ x - y \cdot \left(z \cdot x\right) \]
      cancel-sign-sub-inv [=>] 99.8%	\[ \color{blue}{x + \left(-y\right) \cdot \left(z \cdot x\right)} \]
      flip3-+ [=>] 20.2%	\[ \color{blue}{\frac{{x}^{3} + {\left(\left(-y\right) \cdot \left(z \cdot x\right)\right)}^{3}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)}} \]

   5. Taylor expanded in y around inf 0.1%

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

   6. Simplified 20.2%

      \[\leadsto \frac{\color{blue}{{\left(y \cdot \left(z \cdot \left(-x\right)\right)\right)}^{3}}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 0.1%	\[ \frac{-1 \cdot \left({y}^{3} \cdot \left({z}^{3} \cdot {x}^{3}\right)\right)}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      mul-1-neg [=>] 0.1%	\[ \frac{\color{blue}{-{y}^{3} \cdot \left({z}^{3} \cdot {x}^{3}\right)}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      cube-prod [<=] 8.9%	\[ \frac{-{y}^{3} \cdot \color{blue}{{\left(z \cdot x\right)}^{3}}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      distribute-lft-neg-in [=>] 8.9%	\[ \frac{\color{blue}{\left(-{y}^{3}\right) \cdot {\left(z \cdot x\right)}^{3}}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      cube-neg [<=] 8.9%	\[ \frac{\color{blue}{{\left(-y\right)}^{3}} \cdot {\left(z \cdot x\right)}^{3}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      cube-prod [<=] 20.2%	\[ \frac{\color{blue}{{\left(\left(-y\right) \cdot \left(z \cdot x\right)\right)}^{3}}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      distribute-lft-neg-in [<=] 20.2%	\[ \frac{{\color{blue}{\left(-y \cdot \left(z \cdot x\right)\right)}}^{3}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 20.2%	\[ \frac{{\color{blue}{\left(y \cdot \left(-z \cdot x\right)\right)}}^{3}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]
      distribute-rgt-neg-in [=>] 20.2%	\[ \frac{{\left(y \cdot \color{blue}{\left(z \cdot \left(-x\right)\right)}\right)}^{3}}{x \cdot x + \left(\left(\left(-y\right) \cdot \left(z \cdot x\right)\right) \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right) - x \cdot \left(\left(-y\right) \cdot \left(z \cdot x\right)\right)\right)} \]

   7. Taylor expanded in y around inf 99.8%

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

   8. Simplified 97.4%

      \[\leadsto \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]
      Step-by-step derivation

      [Start] 99.8%	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 99.8%	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 99.8%	\[ -\color{blue}{\left(z \cdot x\right) \cdot y} \]
      associate-*r* [<=] 97.4%	\[ -\color{blue}{z \cdot \left(x \cdot y\right)} \]
      distribute-rgt-neg-in [=>] 97.4%	\[ \color{blue}{z \cdot \left(-x \cdot y\right)} \]
      *-commutative [=>] 97.4%	\[ z \cdot \left(-\color{blue}{y \cdot x}\right) \]
      distribute-rgt-neg-in [=>] 97.4%	\[ z \cdot \color{blue}{\left(y \cdot \left(-x\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+239}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+239}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+168}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	74.6%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+101} \lor \neg \left(y \leq 4.6 \cdot 10^{-115}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	75.4%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \left(-z \cdot x\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \end{array} \]

Alternative 4
Accuracy	75.1%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+101}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \end{array} \]

Alternative 5
Accuracy	50.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))