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

Average Error: 5.21% → 0.94%

Time: 6.0s

Precision: binary64

Cost: 7304

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+161}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, -y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) -5e+161)
   (* z (* y (- x)))
   (if (<= (* y z) 1e+124) (* x (fma z (- y) 1.0)) (* y (* z (- 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) <= -5e+161) {
		tmp = z * (y * -x);
	} else if ((y * z) <= 1e+124) {
		tmp = x * fma(z, -y, 1.0);
	} else {
		tmp = y * (z * -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) <= -5e+161)
		tmp = Float64(z * Float64(y * Float64(-x)));
	elseif (Float64(y * z) <= 1e+124)
		tmp = Float64(x * fma(z, Float64(-y), 1.0));
	else
		tmp = Float64(y * Float64(z * Float64(-x)));
	end
	return 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], -5e+161], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 1e+124], N[(x * N[(z * (-y) + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(z * (-x)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -4.9999999999999997e161

   1. Initial program 30.16

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

   2. Taylor expanded in y around inf 3.32

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

   3. Simplified 4.27

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

      [Start] 3.32	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 3.32	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 3.32	\[ -\color{blue}{\left(z \cdot x\right) \cdot y} \]
      associate-*l* [=>] 4.27	\[ -\color{blue}{z \cdot \left(x \cdot y\right)} \]
      distribute-rgt-neg-in [=>] 4.27	\[ \color{blue}{z \cdot \left(-x \cdot y\right)} \]
      *-commutative [<=] 4.27	\[ z \cdot \left(-\color{blue}{y \cdot x}\right) \]
      distribute-rgt-neg-in [=>] 4.27	\[ z \cdot \color{blue}{\left(y \cdot \left(-x\right)\right)} \]

   if -4.9999999999999997e161 < (*.f64 y z) < 9.99999999999999948e123

   1. Initial program 0.13

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

   2. Simplified 0.13

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

      [Start] 0.13	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 0.13	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 0.13	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 0.13	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 0.13	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

   if 9.99999999999999948e123 < (*.f64 y z)

   1. Initial program 26.83

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

   2. Simplified 26.83

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

      [Start] 26.83	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 26.83	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 26.83	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 26.83	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 26.83	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

   3. Taylor expanded in z around inf 4.9

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

   4. Simplified 4.9

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

      [Start] 4.9	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      associate-*r* [=>] 4.9	\[ \color{blue}{\left(-1 \cdot y\right) \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 4.9	\[ \color{blue}{\left(z \cdot x\right) \cdot \left(-1 \cdot y\right)} \]
      mul-1-neg [=>] 4.9	\[ \left(z \cdot x\right) \cdot \color{blue}{\left(-y\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.94

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+161}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 10^{+124}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, -y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.94%
Cost	968

\[\begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+161}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \cdot z \leq 10^{+124}:\\ \;\;\;\;x \cdot \left(1 - y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 2
Error	27.88%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-74} \lor \neg \left(z \leq 4.3 \cdot 10^{+142}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	25.87%
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-51} \lor \neg \left(z \leq 2.1 \cdot 10^{+139}\right):\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	26.3%
Cost	648

\[\begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-55}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+148}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 5
Error	39.03%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023096 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))