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

Average Error: 5.77% → 0.58%

Time: 8.4s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+148}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 1.0 y) z)))
   (if (<= t_0 -4e+148)
     (* z (* x (+ y -1.0)))
     (if (<= t_0 5e+131) (* x (- 1.0 t_0)) (* z (- (* y x) x))))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -4e+148) {
		tmp = z * (x * (y + -1.0));
	} else if (t_0 <= 5e+131) {
		tmp = x * (1.0 - t_0);
	} else {
		tmp = z * ((y * x) - 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 - ((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) :: tmp
    t_0 = (1.0d0 - y) * z
    if (t_0 <= (-4d+148)) then
        tmp = z * (x * (y + (-1.0d0)))
    else if (t_0 <= 5d+131) then
        tmp = x * (1.0d0 - t_0)
    else
        tmp = z * ((y * x) - x)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (1.0 - y) * z;
	double tmp;
	if (t_0 <= -4e+148) {
		tmp = z * (x * (y + -1.0));
	} else if (t_0 <= 5e+131) {
		tmp = x * (1.0 - t_0);
	} else {
		tmp = z * ((y * x) - x);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (1.0 - y) * z
	tmp = 0
	if t_0 <= -4e+148:
		tmp = z * (x * (y + -1.0))
	elif t_0 <= 5e+131:
		tmp = x * (1.0 - t_0)
	else:
		tmp = z * ((y * x) - x)
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(1.0 - y) * z)
	tmp = 0.0
	if (t_0 <= -4e+148)
		tmp = Float64(z * Float64(x * Float64(y + -1.0)));
	elseif (t_0 <= 5e+131)
		tmp = Float64(x * Float64(1.0 - t_0));
	else
		tmp = Float64(z * Float64(Float64(y * x) - x));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (1.0 - ((1.0 - y) * z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (1.0 - y) * z;
	tmp = 0.0;
	if (t_0 <= -4e+148)
		tmp = z * (x * (y + -1.0));
	elseif (t_0 <= 5e+131)
		tmp = x * (1.0 - t_0);
	else
		tmp = z * ((y * x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+148], N[(z * N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+131], N[(x * N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	5.77%
Target	0.42%
Herbie	0.58%

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \mathbf{elif}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) < 3.892237649663903 \cdot 10^{+134}:\\ \;\;\;\;\left(x \cdot y\right) \cdot z - \left(x \cdot z - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - y\right) \cdot \left(\left(-z\right) \cdot x\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 1 y) z) < -4.0000000000000002e148

   1. Initial program 22.35

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

   2. Taylor expanded in z around inf 1.8

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

   if -4.0000000000000002e148 < (*.f64 (-.f64 1 y) z) < 4.99999999999999995e131

   1. Initial program 0.1

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

   if 4.99999999999999995e131 < (*.f64 (-.f64 1 y) z)

   1. Initial program 18.99

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

   2. Taylor expanded in x around 0 18.99

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

   3. Simplified 1.87

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

      [Start] 18.99	\[ \left(1 - z \cdot \left(1 - y\right)\right) \cdot x \]
      distribute-rgt-out-- [<=] 19	\[ \left(1 - \color{blue}{\left(1 \cdot z - y \cdot z\right)}\right) \cdot x \]
      *-lft-identity [=>] 19	\[ \left(1 - \left(\color{blue}{z} - y \cdot z\right)\right) \cdot x \]
      associate-+l- [<=] 19	\[ \color{blue}{\left(\left(1 - z\right) + y \cdot z\right)} \cdot x \]
      +-commutative [=>] 19	\[ \color{blue}{\left(y \cdot z + \left(1 - z\right)\right)} \cdot x \]
      sub-neg [=>] 19	\[ \left(y \cdot z + \color{blue}{\left(1 + \left(-z\right)\right)}\right) \cdot x \]
      +-commutative [=>] 19	\[ \left(y \cdot z + \color{blue}{\left(\left(-z\right) + 1\right)}\right) \cdot x \]
      mul-1-neg [<=] 19	\[ \left(y \cdot z + \left(\color{blue}{-1 \cdot z} + 1\right)\right) \cdot x \]
      associate-+r+ [=>] 19	\[ \color{blue}{\left(\left(y \cdot z + -1 \cdot z\right) + 1\right)} \cdot x \]
      distribute-rgt-in [<=] 18.99	\[ \left(\color{blue}{z \cdot \left(y + -1\right)} + 1\right) \cdot x \]
      distribute-lft1-in [<=] 18.99	\[ \color{blue}{\left(z \cdot \left(y + -1\right)\right) \cdot x + x} \]
      metadata-eval [<=] 18.99	\[ \left(z \cdot \left(y + \color{blue}{\left(-1\right)}\right)\right) \cdot x + x \]
      sub-neg [<=] 18.99	\[ \left(z \cdot \color{blue}{\left(y - 1\right)}\right) \cdot x + x \]
      associate-*r* [<=] 1.87	\[ \color{blue}{z \cdot \left(\left(y - 1\right) \cdot x\right)} + x \]
      fma-def [=>] 1.87	\[ \color{blue}{\mathsf{fma}\left(z, \left(y - 1\right) \cdot x, x\right)} \]
      *-commutative [=>] 1.87	\[ \mathsf{fma}\left(z, \color{blue}{x \cdot \left(y - 1\right)}, x\right) \]
      sub-neg [=>] 1.87	\[ \mathsf{fma}\left(z, x \cdot \color{blue}{\left(y + \left(-1\right)\right)}, x\right) \]
      metadata-eval [=>] 1.87	\[ \mathsf{fma}\left(z, x \cdot \left(y + \color{blue}{-1}\right), x\right) \]

   4. Taylor expanded in z around inf 1.87

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

   5. Simplified 1.86

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

      [Start] 1.87	\[ z \cdot \left(\left(y - 1\right) \cdot x\right) \]
      *-commutative [=>] 1.87	\[ z \cdot \color{blue}{\left(x \cdot \left(y - 1\right)\right)} \]
      distribute-rgt-out-- [<=] 1.86	\[ z \cdot \color{blue}{\left(y \cdot x - 1 \cdot x\right)} \]
      *-commutative [<=] 1.86	\[ z \cdot \left(\color{blue}{x \cdot y} - 1 \cdot x\right) \]
      *-lft-identity [=>] 1.86	\[ z \cdot \left(x \cdot y - \color{blue}{x}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.58

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) \cdot z \leq -4 \cdot 10^{+148}:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;\left(1 - y\right) \cdot z \leq 5 \cdot 10^{+131}:\\ \;\;\;\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	13.87%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -80000000 \lor \neg \left(z \leq 14.8\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \end{array} \]

Alternative 2
Error	13.93%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -92000000:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;z \leq 0.44:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternative 3
Error	1.41%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -0.95:\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x - x\right)\\ \end{array} \]

Alternative 4
Error	17.66%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+36}:\\ \;\;\;\;y \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+114}:\\ \;\;\;\;x \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 5
Error	29.94%
Cost	521

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

Alternative 6
Error	52.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, J"
  :precision binary64

  :herbie-target
  (if (< (* x (- 1.0 (* (- 1.0 y) z))) -1.618195973607049e+50) (+ x (* (- 1.0 y) (* (- z) x))) (if (< (* x (- 1.0 (* (- 1.0 y) z))) 3.892237649663903e+134) (- (* (* x y) z) (- (* x z) x)) (+ x (* (- 1.0 y) (* (- z) x)))))

  (* x (- 1.0 (* (- 1.0 y) z))))