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

Average Error: 3.4 → 1.0

Time: 10.6s

Precision: binary64

Cost: 2632

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(1 - y \cdot z\right)\\ t_1 := x - z \cdot \left(y \cdot x\right)\\ t_2 := t_1 \cdot \left(t_1 \cdot \frac{1}{t_1}\right)\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 10^{+300}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* y z))))
        (t_1 (- x (* z (* y x))))
        (t_2 (* t_1 (* t_1 (/ 1.0 t_1)))))
   (if (<= t_0 -2e+262) t_2 (if (<= t_0 1e+300) (- x (* x (* y z))) t_2))))

C

double code(double x, double y, double z) {
	return x * (1.0 - (y * z));
}

↓

double code(double x, double y, double z) {
	double t_0 = x * (1.0 - (y * z));
	double t_1 = x - (z * (y * x));
	double t_2 = t_1 * (t_1 * (1.0 / t_1));
	double tmp;
	if (t_0 <= -2e+262) {
		tmp = t_2;
	} else if (t_0 <= 1e+300) {
		tmp = x - (x * (y * z));
	} else {
		tmp = t_2;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (1.0d0 - (y * z))
    t_1 = x - (z * (y * x))
    t_2 = t_1 * (t_1 * (1.0d0 / t_1))
    if (t_0 <= (-2d+262)) then
        tmp = t_2
    else if (t_0 <= 1d+300) then
        tmp = x - (x * (y * z))
    else
        tmp = t_2
    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 t_0 = x * (1.0 - (y * z));
	double t_1 = x - (z * (y * x));
	double t_2 = t_1 * (t_1 * (1.0 / t_1));
	double tmp;
	if (t_0 <= -2e+262) {
		tmp = t_2;
	} else if (t_0 <= 1e+300) {
		tmp = x - (x * (y * z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	return x * (1.0 - (y * z))

↓

def code(x, y, z):
	t_0 = x * (1.0 - (y * z))
	t_1 = x - (z * (y * x))
	t_2 = t_1 * (t_1 * (1.0 / t_1))
	tmp = 0
	if t_0 <= -2e+262:
		tmp = t_2
	elif t_0 <= 1e+300:
		tmp = x - (x * (y * z))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	return Float64(x * Float64(1.0 - Float64(y * z)))
end

↓

function code(x, y, z)
	t_0 = Float64(x * Float64(1.0 - Float64(y * z)))
	t_1 = Float64(x - Float64(z * Float64(y * x)))
	t_2 = Float64(t_1 * Float64(t_1 * Float64(1.0 / t_1)))
	tmp = 0.0
	if (t_0 <= -2e+262)
		tmp = t_2;
	elseif (t_0 <= 1e+300)
		tmp = Float64(x - Float64(x * Float64(y * z)));
	else
		tmp = t_2;
	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)
	t_0 = x * (1.0 - (y * z));
	t_1 = x - (z * (y * x));
	t_2 = t_1 * (t_1 * (1.0 / t_1));
	tmp = 0.0;
	if (t_0 <= -2e+262)
		tmp = t_2;
	elseif (t_0 <= 1e+300)
		tmp = x - (x * (y * z));
	else
		tmp = t_2;
	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_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[(z * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+262], t$95$2, If[LessEqual[t$95$0, 1e+300], N[(x - N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 1 (*.f64 y z))) < -2e262 or 1.0000000000000001e300 < (*.f64 x (-.f64 1 (*.f64 y z)))

   1. Initial program 32.4

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

   2. Simplified 32.4

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

      [Start] 32.4	\[ x \cdot \left(1 - y \cdot z\right) \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 32.4	\[ \color{blue}{1 \cdot x - x \cdot \left(y \cdot z\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 32.4	\[ \color{blue}{x \cdot 1} - x \cdot \left(y \cdot z\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 32.4	\[ \color{blue}{x} - x \cdot \left(y \cdot z\right) \]

   3. Taylor expanded in x around 0 7.2

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

   4. Simplified 8.8

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

      [Start] 7.2	\[ x - y \cdot \left(z \cdot x\right) \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 8.8	\[ x - \color{blue}{z \cdot \left(y \cdot x\right)} \]

   5. Applied egg-rr 8.8

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

   if -2e262 < (*.f64 x (-.f64 1 (*.f64 y z))) < 1.0000000000000001e300

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ x \cdot \left(1 - y \cdot z\right) \]
      rational_best_oopsla_all_46_json_45_simplify-13 [=>] 0.1	\[ \color{blue}{1 \cdot x - x \cdot \left(y \cdot z\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \color{blue}{x \cdot 1} - x \cdot \left(y \cdot z\right) \]
      rational_best_oopsla_all_46_json_45_simplify-52 [=>] 0.1	\[ \color{blue}{x} - x \cdot \left(y \cdot z\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - y \cdot z\right) \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\left(x - z \cdot \left(y \cdot x\right)\right) \cdot \left(\left(x - z \cdot \left(y \cdot x\right)\right) \cdot \frac{1}{x - z \cdot \left(y \cdot x\right)}\right)\\ \mathbf{elif}\;x \cdot \left(1 - y \cdot z\right) \leq 10^{+300}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - z \cdot \left(y \cdot x\right)\right) \cdot \left(\left(x - z \cdot \left(y \cdot x\right)\right) \cdot \frac{1}{x - z \cdot \left(y \cdot x\right)}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	1480

\[\begin{array}{l} t_0 := x \cdot \left(1 - y \cdot z\right)\\ t_1 := z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	1480

\[\begin{array}{l} t_0 := x \cdot \left(1 - y \cdot z\right)\\ t_1 := z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.3
Cost	1480

\[\begin{array}{l} t_0 := x \cdot \left(1 - y \cdot z\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;t_0 \leq 10^{+300}:\\ \;\;\;\;x - x \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4
Error	17.9
Cost	912

\[\begin{array}{l} t_0 := y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-123}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	4.8
Cost	904

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -20000:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 0.04:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 6
Error	26.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))