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

Average Accuracy: 95.0% → 99.4%

Time: 8.7s

Precision: binary64

Cost: 968

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\frac{-1}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+169}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -1e+181)
   (/ 1.0 (/ -1.0 (* y (* z x))))
   (if (<= (* y z) 5e+169) (- 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+181) {
		tmp = 1.0 / (-1.0 / (y * (z * x)));
	} else if ((y * z) <= 5e+169) {
		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+181)) then
        tmp = 1.0d0 / ((-1.0d0) / (y * (z * x)))
    else if ((y * z) <= 5d+169) 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+181) {
		tmp = 1.0 / (-1.0 / (y * (z * x)));
	} else if ((y * z) <= 5e+169) {
		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+181:
		tmp = 1.0 / (-1.0 / (y * (z * x)))
	elif (y * z) <= 5e+169:
		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+181)
		tmp = Float64(1.0 / Float64(-1.0 / Float64(y * Float64(z * x))));
	elseif (Float64(y * z) <= 5e+169)
		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+181)
		tmp = 1.0 / (-1.0 / (y * (z * x)));
	elseif ((y * z) <= 5e+169)
		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+181], N[(1.0 / N[(-1.0 / N[(y * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e+169], N[(x - N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -9.9999999999999992e180

   1. Initial program 62.8%

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

   2. Taylor expanded in x around 0 62.8%

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

   3. Simplified 97.0%

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

      [Start] 62.8	\[ \left(1 - y \cdot z\right) \cdot x \]
      *-commutative [<=] 62.8	\[ \color{blue}{x \cdot \left(1 - y \cdot z\right)} \]
      distribute-rgt-out-- [<=] 62.8	\[ \color{blue}{1 \cdot x - \left(y \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 62.8	\[ \color{blue}{x} - \left(y \cdot z\right) \cdot x \]
      associate-*r* [<=] 97.0	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]

   4. Applied egg-rr 48.9%

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

      [Start] 97.0	\[ x - y \cdot \left(z \cdot x\right) \]
      flip-- [=>] 49.1	\[ \color{blue}{\frac{x \cdot x - \left(y \cdot \left(z \cdot x\right)\right) \cdot \left(y \cdot \left(z \cdot x\right)\right)}{x + y \cdot \left(z \cdot x\right)}} \]
      clear-num [=>] 48.9	\[ \color{blue}{\frac{1}{\frac{x + y \cdot \left(z \cdot x\right)}{x \cdot x - \left(y \cdot \left(z \cdot x\right)\right) \cdot \left(y \cdot \left(z \cdot x\right)\right)}}} \]
      pow2 [=>] 48.9	\[ \frac{1}{\frac{x + y \cdot \left(z \cdot x\right)}{x \cdot x - \color{blue}{{\left(y \cdot \left(z \cdot x\right)\right)}^{2}}}} \]

   5. Taylor expanded in y around inf 96.8%

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

   if -9.9999999999999992e180 < (*.f64 y z) < 5.00000000000000017e169

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

   3. Simplified 91.7%

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

      [Start] 99.9	\[ \left(1 - y \cdot z\right) \cdot x \]
      *-commutative [<=] 99.9	\[ \color{blue}{x \cdot \left(1 - y \cdot z\right)} \]
      distribute-rgt-out-- [<=] 99.9	\[ \color{blue}{1 \cdot x - \left(y \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 99.9	\[ \color{blue}{x} - \left(y \cdot z\right) \cdot x \]
      associate-*r* [<=] 91.7	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]

   4. Applied egg-rr 68.5%

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

      [Start] 91.7	\[ x - y \cdot \left(z \cdot x\right) \]
      expm1-log1p-u [=>] 78.0	\[ x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)\right)} \]
      expm1-udef [=>] 68.5	\[ x - \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)} - 1\right)} \]

   5. Simplified 99.9%

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

      [Start] 68.5	\[ x - \left(e^{\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)} - 1\right) \]
      expm1-def [=>] 78.0	\[ x - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \left(z \cdot x\right)\right)\right)} \]
      expm1-log1p [=>] 91.7	\[ x - \color{blue}{y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 99.9	\[ x - \color{blue}{\left(y \cdot z\right) \cdot x} \]

   if 5.00000000000000017e169 < (*.f64 y z)

   1. Initial program 67.0%

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

   2. Taylor expanded in y around inf 97.2%

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

   3. Simplified 96.1%

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

      [Start] 97.2	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 97.2	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      associate-*r* [=>] 67.0	\[ -\color{blue}{\left(y \cdot z\right) \cdot x} \]
      distribute-rgt-neg-in [=>] 67.0	\[ \color{blue}{\left(y \cdot z\right) \cdot \left(-x\right)} \]
      *-commutative [=>] 67.0	\[ \color{blue}{\left(z \cdot y\right)} \cdot \left(-x\right) \]
      associate-*l* [=>] 96.1	\[ \color{blue}{z \cdot \left(y \cdot \left(-x\right)\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+181}:\\ \;\;\;\;\frac{1}{\frac{-1}{y \cdot \left(z \cdot x\right)}}\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+169}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+134}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+169}:\\ \;\;\;\;x - \left(y \cdot z\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	70.1%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -2.06 \cdot 10^{-156} \lor \neg \left(z \leq 10^{+63}\right):\\ \;\;\;\;x \cdot \left(z \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	73.1%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-80} \lor \neg \left(z \leq 1.36 \cdot 10^{+63}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	60.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023143 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))