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

Average Accuracy: 94.5% → 97.1%

Time: 5.6s

Precision: binary64

Cost: 708

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq 10^{+302}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) 1e+302) (- x (* (* y z) x)) (* 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) <= 1e+302) {
		tmp = x - ((y * z) * x);
	} 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) <= 1d+302) then
        tmp = x - ((y * z) * x)
    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) <= 1e+302) {
		tmp = x - ((y * z) * x);
	} 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) <= 1e+302:
		tmp = x - ((y * z) * x)
	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) <= 1e+302)
		tmp = Float64(x - Float64(Float64(y * z) * x));
	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) <= 1e+302)
		tmp = x - ((y * z) * x);
	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], 1e+302], N[(x - N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < 1.0000000000000001e302

   1. Initial program 97.0%

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

   2. Taylor expanded in x around 0 97.0%

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

   3. Simplified 92.5%

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

      [Start] 97.0	\[ \left(1 - y \cdot z\right) \cdot x \]
      *-commutative [<=] 97.0	\[ \color{blue}{x \cdot \left(1 - y \cdot z\right)} \]
      distribute-rgt-out-- [<=] 97.0	\[ \color{blue}{1 \cdot x - \left(y \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 97.0	\[ \color{blue}{x} - \left(y \cdot z\right) \cdot x \]
      associate-*r* [<=] 92.5	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]

   4. Applied egg-rr 52.4%

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

   5. Simplified 96.8%

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

      [Start] 52.4	\[ \left(x \cdot x - {\left(x \cdot \left(y \cdot z\right)\right)}^{2}\right) \cdot \frac{1}{x + x \cdot \left(y \cdot z\right)} \]
      *-commutative [=>] 52.4	\[ \color{blue}{\frac{1}{x + x \cdot \left(y \cdot z\right)} \cdot \left(x \cdot x - {\left(x \cdot \left(y \cdot z\right)\right)}^{2}\right)} \]
      associate-/r/ [<=] 52.4	\[ \color{blue}{\frac{1}{\frac{x + x \cdot \left(y \cdot z\right)}{x \cdot x - {\left(x \cdot \left(y \cdot z\right)\right)}^{2}}}} \]
      unpow2 [=>] 52.4	\[ \frac{1}{\frac{x + x \cdot \left(y \cdot z\right)}{x \cdot x - \color{blue}{\left(x \cdot \left(y \cdot z\right)\right) \cdot \left(x \cdot \left(y \cdot z\right)\right)}}} \]
      difference-of-squares [=>] 52.4	\[ \frac{1}{\frac{x + x \cdot \left(y \cdot z\right)}{\color{blue}{\left(x + x \cdot \left(y \cdot z\right)\right) \cdot \left(x - x \cdot \left(y \cdot z\right)\right)}}} \]
      associate-/r* [=>] 96.8	\[ \frac{1}{\color{blue}{\frac{\frac{x + x \cdot \left(y \cdot z\right)}{x + x \cdot \left(y \cdot z\right)}}{x - x \cdot \left(y \cdot z\right)}}} \]

   6. Applied egg-rr 97.0%

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

   if 1.0000000000000001e302 < (*.f64 y z)

   1. Initial program 7.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   3. Simplified 99.5%

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

      [Start] 99.6	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 99.6	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 7.6	\[ -\color{blue}{\left(y \cdot z\right) \cdot x} \]
      distribute-rgt-neg-in [=>] 7.6	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]
      *-commutative [=>] 7.6	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(-x\right) \]
      associate-*l* [=>] 99.5	\[ \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq 10^{+302}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	70.1%
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+155} \lor \neg \left(y \leq -6 \cdot 10^{+93} \lor \neg \left(y \leq -2.25 \cdot 10^{+45}\right) \land y \leq 2.05 \cdot 10^{-78}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	72.8%
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+153} \lor \neg \left(y \leq -1.5 \cdot 10^{+96} \lor \neg \left(y \leq -2.1 \cdot 10^{+45}\right) \land y \leq 2.3 \cdot 10^{-72}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	97.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq 10^{+302}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	60.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023129 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))