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

Average Accuracy: 95.1% → 99.4%

Time: 10.2s

Precision: binary64

Cost: 7304

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y z) (- INFINITY))
   (* z (* y (- x)))
   (if (<= (* y z) 5e+148) (* 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) <= -((double) INFINITY)) {
		tmp = z * (y * -x);
	} else if ((y * z) <= 5e+148) {
		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) <= Float64(-Inf))
		tmp = Float64(z * Float64(y * Float64(-x)));
	elseif (Float64(y * z) <= 5e+148)
		tmp = Float64(x * fma(z, Float64(-y), 1.0));
	else
		tmp = Float64(Float64(-y) * Float64(z * 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], (-Infinity)], N[(z * N[(y * (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e+148], 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) < -inf.0

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf 99.7%

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

   3. Simplified 99.7%

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

      [Start] 99.7	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      mul-1-neg [=>] 99.7	\[ \color{blue}{-y \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 99.7	\[ -\color{blue}{\left(z \cdot x\right) \cdot y} \]
      associate-*l* [=>] 99.7	\[ -\color{blue}{z \cdot \left(x \cdot y\right)} \]
      distribute-rgt-neg-in [=>] 99.7	\[ \color{blue}{z \cdot \left(-x \cdot y\right)} \]
      *-commutative [<=] 99.7	\[ z \cdot \left(-\color{blue}{y \cdot x}\right) \]
      distribute-rgt-neg-in [=>] 99.7	\[ z \cdot \color{blue}{\left(y \cdot \left(-x\right)\right)} \]

   if -inf.0 < (*.f64 y z) < 5.00000000000000024e148

   1. Initial program 99.9%

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

   2. Simplified 99.9%

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

      [Start] 99.9	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 99.9	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 99.9	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 99.9	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 99.9	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

   if 5.00000000000000024e148 < (*.f64 y z)

   1. Initial program 73.2%

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

   2. Simplified 73.2%

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

      [Start] 73.2	\[ x \cdot \left(1 - y \cdot z\right) \]
      cancel-sign-sub-inv [=>] 73.2	\[ x \cdot \color{blue}{\left(1 + \left(-y\right) \cdot z\right)} \]
      +-commutative [=>] 73.2	\[ x \cdot \color{blue}{\left(\left(-y\right) \cdot z + 1\right)} \]
      *-commutative [=>] 73.2	\[ x \cdot \left(\color{blue}{z \cdot \left(-y\right)} + 1\right) \]
      fma-def [=>] 73.2	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -y, 1\right)} \]

   3. Taylor expanded in z around inf 94.5%

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

   4. Simplified 94.5%

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

      [Start] 94.5	\[ -1 \cdot \left(y \cdot \left(z \cdot x\right)\right) \]
      associate-*r* [=>] 94.5	\[ \color{blue}{\left(-1 \cdot y\right) \cdot \left(z \cdot x\right)} \]
      *-commutative [=>] 94.5	\[ \color{blue}{\left(z \cdot x\right) \cdot \left(-1 \cdot y\right)} \]
      mul-1-neg [=>] 94.5	\[ \left(z \cdot x\right) \cdot \color{blue}{\left(-y\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.8%
Cost	1224

\[\begin{array}{l} t_0 := 1 - y \cdot z\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+285}:\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	968

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

Alternative 3
Accuracy	70.3%
Cost	914

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+136} \lor \neg \left(y \leq -1.95 \cdot 10^{+73}\right) \land \left(y \leq -6.8 \cdot 10^{+47} \lor \neg \left(y \leq 9 \cdot 10^{-20}\right)\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	72.6%
Cost	912

\[\begin{array}{l} t_0 := z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{if}\;y \leq -9.8 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4 \cdot 10^{+72}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{+48}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	72.3%
Cost	912

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+133}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-x\right)\right)\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{+73}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{+46}:\\ \;\;\;\;\left(y \cdot z\right) \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{-21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z \cdot x\right)\\ \end{array} \]

Alternative 6
Accuracy	60.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x y z)
  :name "Data.Colour.RGBSpace.HSV:hsv from colour-2.3.3, I"
  :precision binary64
  (* x (- 1.0 (* y z))))