Data.Colour.RGB:hslsv from colour-2.3.3, C

Average Error: 0.03% → 0.04%

Time: 8.2s

Precision: binary64

Cost: 2112

Math

\[\frac{x - y}{2 - \left(x + y\right)} \]

↓

\[\begin{array}{l} t_0 := \frac{x - y}{\left(2 - y\right) - x}\\ t_1 := 2 + t_0\\ \frac{t_0 \cdot t_1}{t_1} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (- x y) (- 2.0 (+ x y))))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- (- 2.0 y) x))) (t_1 (+ 2.0 t_0)))
   (/ (* t_0 t_1) t_1)))

C

double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

↓

double code(double x, double y) {
	double t_0 = (x - y) / ((2.0 - y) - x);
	double t_1 = 2.0 + t_0;
	return (t_0 * t_1) / t_1;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) / (2.0d0 - (x + y))
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (x - y) / ((2.0d0 - y) - x)
    t_1 = 2.0d0 + t_0
    code = (t_0 * t_1) / t_1
end function

Java

public static double code(double x, double y) {
	return (x - y) / (2.0 - (x + y));
}

↓

public static double code(double x, double y) {
	double t_0 = (x - y) / ((2.0 - y) - x);
	double t_1 = 2.0 + t_0;
	return (t_0 * t_1) / t_1;
}

Python

def code(x, y):
	return (x - y) / (2.0 - (x + y))

↓

def code(x, y):
	t_0 = (x - y) / ((2.0 - y) - x)
	t_1 = 2.0 + t_0
	return (t_0 * t_1) / t_1

Julia

function code(x, y)
	return Float64(Float64(x - y) / Float64(2.0 - Float64(x + y)))
end

↓

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(2.0 - y) - x))
	t_1 = Float64(2.0 + t_0)
	return Float64(Float64(t_0 * t_1) / t_1)
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) / (2.0 - (x + y));
end

↓

function tmp = code(x, y)
	t_0 = (x - y) / ((2.0 - y) - x);
	t_1 = 2.0 + t_0;
	tmp = (t_0 * t_1) / t_1;
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(2.0 - N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(N[(2.0 - y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, N[(N[(t$95$0 * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Target

Original	0.03%
Target	0.03%
Herbie	0.04%

\[\frac{x}{2 - \left(x + y\right)} - \frac{y}{2 - \left(x + y\right)} \]

Derivation

1. Initial program 0.03

   \[\frac{x - y}{2 - \left(x + y\right)} \]

2. Simplified 0.03

   \[\leadsto \color{blue}{\frac{x - y}{\left(2 - x\right) - y}} \]
   Proof

   [Start] 0.03	\[ \frac{x - y}{2 - \left(x + y\right)} \]
   associate--r+ [=>] 0.03	\[ \frac{x - y}{\color{blue}{\left(2 - x\right) - y}} \]

3. Applied egg-rr 1.34

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x - y}{2 - \left(x + y\right)}\right)\right)} \]

4. Applied egg-rr 21.74

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

5. Simplified 0.04

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

   [Start] 21.74	\[ \frac{\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) \cdot \left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) - 1}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   difference-of-sqr-1 [=>] 21.74	\[ \frac{\color{blue}{\left(\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) + 1\right) \cdot \left(\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) - 1\right)}}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   associate--l+ [=>] 0.05	\[ \frac{\left(\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) + 1\right) \cdot \color{blue}{\left(\frac{x - y}{\left(2 - y\right) - x} + \left(1 - 1\right)\right)}}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   metadata-eval [=>] 0.05	\[ \frac{\left(\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) + 1\right) \cdot \left(\frac{x - y}{\left(2 - y\right) - x} + \color{blue}{0}\right)}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   +-rgt-identity [=>] 0.05	\[ \frac{\left(\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) + 1\right) \cdot \color{blue}{\frac{x - y}{\left(2 - y\right) - x}}}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   *-commutative [=>] 0.05	\[ \frac{\color{blue}{\frac{x - y}{\left(2 - y\right) - x} \cdot \left(\left(\frac{x - y}{\left(2 - y\right) - x} + 1\right) + 1\right)}}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   associate-+l+ [=>] 0.04	\[ \frac{\frac{x - y}{\left(2 - y\right) - x} \cdot \color{blue}{\left(\frac{x - y}{\left(2 - y\right) - x} + \left(1 + 1\right)\right)}}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   metadata-eval [=>] 0.04	\[ \frac{\frac{x - y}{\left(2 - y\right) - x} \cdot \left(\frac{x - y}{\left(2 - y\right) - x} + \color{blue}{2}\right)}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]
   +-commutative [<=] 0.04	\[ \frac{\frac{x - y}{\left(2 - y\right) - x} \cdot \color{blue}{\left(2 + \frac{x - y}{\left(2 - y\right) - x}\right)}}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]

6. Final simplification 0.04

   \[\leadsto \frac{\frac{x - y}{\left(2 - y\right) - x} \cdot \left(2 + \frac{x - y}{\left(2 - y\right) - x}\right)}{2 + \frac{x - y}{\left(2 - y\right) - x}} \]

Alternatives

Alternative 1
Error	26.46%
Cost	976

\[\begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{-71}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{y}{x}} + 1\\ \end{array} \]

Alternative 2
Error	25.49%
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-71}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	26.61%
Cost	848

\[\begin{array}{l} t_0 := \frac{x}{2 - x}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{y + -2}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-71}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	38.23%
Cost	724

\[\begin{array}{l} \mathbf{if}\;y \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-118}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-89}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-71}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	0.03%
Cost	576

\[\frac{x - y}{2 - \left(x + y\right)} \]

Alternative 6
Error	36.84%
Cost	328

\[\begin{array}{l} \mathbf{if}\;y \leq -11500000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{+60}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	62.54%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x y)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, C"
  :precision binary64

  :herbie-target
  (- (/ x (- 2.0 (+ x y))) (/ y (- 2.0 (+ x y))))

  (/ (- x y) (- 2.0 (+ x y))))