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

Percentage Accurate: 95.9% → 96.8%

Time: 7.1s

Precision: binary64

Cost: 580

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \left(x \cdot y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.5e-14) (* x (- 1.0 (* z y))) (- x (* 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 (z <= 1.5e-14) {
		tmp = x * (1.0 - (z * y));
	} else {
		tmp = x - (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 (z <= 1.5d-14) then
        tmp = x * (1.0d0 - (z * y))
    else
        tmp = x - (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 (z <= 1.5e-14) {
		tmp = x * (1.0 - (z * y));
	} else {
		tmp = x - (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 z <= 1.5e-14:
		tmp = x * (1.0 - (z * y))
	else:
		tmp = x - (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 (z <= 1.5e-14)
		tmp = Float64(x * Float64(1.0 - Float64(z * y)));
	else
		tmp = Float64(x - Float64(z * Float64(x * 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 (z <= 1.5e-14)
		tmp = x * (1.0 - (z * y));
	else
		tmp = x - (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[z, 1.5e-14], N[(x * N[(1.0 - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(z * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4999999999999999e-14

   1. Initial program 98.4%

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

   if 1.4999999999999999e-14 < z

   1. Initial program 90.9%

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

   2. Applied egg-rr 56.3%

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

      [Start] 90.9%	\[ x \cdot \left(1 - y \cdot z\right) \]
      flip-- [=>] 58.8%	\[ x \cdot \color{blue}{\frac{1 \cdot 1 - \left(y \cdot z\right) \cdot \left(y \cdot z\right)}{1 + y \cdot z}} \]
      associate-*r/ [=>] 56.3%	\[ \color{blue}{\frac{x \cdot \left(1 \cdot 1 - \left(y \cdot z\right) \cdot \left(y \cdot z\right)\right)}{1 + y \cdot z}} \]
      metadata-eval [=>] 56.3%	\[ \frac{x \cdot \left(\color{blue}{1} - \left(y \cdot z\right) \cdot \left(y \cdot z\right)\right)}{1 + y \cdot z} \]
      pow2 [=>] 56.3%	\[ \frac{x \cdot \left(1 - \color{blue}{{\left(y \cdot z\right)}^{2}}\right)}{1 + y \cdot z} \]

   3. Simplified 58.8%

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

      [Start] 56.3%	\[ \frac{x \cdot \left(1 - {\left(y \cdot z\right)}^{2}\right)}{1 + y \cdot z} \]
      associate-/l* [=>] 58.8%	\[ \color{blue}{\frac{x}{\frac{1 + y \cdot z}{1 - {\left(y \cdot z\right)}^{2}}}} \]

   4. Taylor expanded in y around inf 43.7%

      \[\leadsto \frac{x}{\color{blue}{-\left(\frac{1}{{y}^{3} \cdot {z}^{3}} + \left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)\right)}} \]

   5. Simplified 70.1%

      \[\leadsto \frac{x}{\color{blue}{\frac{-1}{{\left(y \cdot z\right)}^{3}} - \left(\frac{1}{{\left(y \cdot z\right)}^{2}} + \left(\frac{\frac{1}{y}}{z} + \frac{1}{{\left(y \cdot z\right)}^{4}}\right)\right)}} \]
      Step-by-step derivation

      [Start] 43.7%	\[ \frac{x}{-\left(\frac{1}{{y}^{3} \cdot {z}^{3}} + \left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)\right)} \]
      distribute-neg-in [=>] 53.4%	\[ \frac{x}{\color{blue}{\left(-\frac{1}{{y}^{3} \cdot {z}^{3}}\right) + \left(-\left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)\right)}} \]
      unsub-neg [=>] 53.4%	\[ \frac{x}{\color{blue}{\left(-\frac{1}{{y}^{3} \cdot {z}^{3}}\right) - \left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)}} \]
      cube-prod [<=] 53.5%	\[ \frac{x}{\left(-\frac{1}{\color{blue}{{\left(y \cdot z\right)}^{3}}}\right) - \left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)} \]
      distribute-neg-frac [=>] 53.5%	\[ \frac{x}{\color{blue}{\frac{-1}{{\left(y \cdot z\right)}^{3}}} - \left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)} \]
      metadata-eval [=>] 53.5%	\[ \frac{x}{\frac{\color{blue}{-1}}{{\left(y \cdot z\right)}^{3}} - \left(\frac{1}{{y}^{2} \cdot {z}^{2}} + \left(\frac{1}{{y}^{4} \cdot {z}^{4}} + \frac{1}{y \cdot z}\right)\right)} \]

   6. Taylor expanded in y around inf 88.3%

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

   7. Simplified 96.1%

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

      [Start] 88.3%	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) + x \]
      +-commutative [=>] 88.3%	\[ \color{blue}{x + -1 \cdot \left(y \cdot \left(z \cdot x\right)\right)} \]
      mul-1-neg [=>] 88.3%	\[ x + \color{blue}{\left(-y \cdot \left(z \cdot x\right)\right)} \]
      unsub-neg [=>] 88.3%	\[ \color{blue}{x - y \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 88.3%	\[ x - \color{blue}{\left(z \cdot x\right) \cdot y} \]
      associate-*l* [=>] 96.1%	\[ x - \color{blue}{z \cdot \left(x \cdot y\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \left(1 - z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	708

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

Alternative 3
Accuracy	73.0%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+14} \lor \neg \left(y \leq 5.2 \cdot 10^{-74}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	75.8%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+45} \lor \neg \left(y \leq 5.5 \cdot 10^{-75}\right):\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	75.0%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -2900000000:\\ \;\;\;\;\left(-z\right) \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{-73}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \left(-z\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	50.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023245 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))