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

Percentage Accurate: 96.2% → 97.7%

Time: 11.2s

Precision: binary64

Cost: 7492

• 21.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 100000:\\ \;\;\;\;x - z \cdot \left(x \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* x (- 1.0 (* (- 1.0 y) z))) 100000.0)
   (- x (* z (* x (- 1.0 y))))
   (fma (+ y -1.0) (* x z) 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 tmp;
	if ((x * (1.0 - ((1.0 - y) * z))) <= 100000.0) {
		tmp = x - (z * (x * (1.0 - y)));
	} else {
		tmp = fma((y + -1.0), (x * z), 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)
	tmp = 0.0
	if (Float64(x * Float64(1.0 - Float64(Float64(1.0 - y) * z))) <= 100000.0)
		tmp = Float64(x - Float64(z * Float64(x * Float64(1.0 - y))));
	else
		tmp = fma(Float64(y + -1.0), Float64(x * z), x);
	end
	return 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_] := If[LessEqual[N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 100000.0], N[(x - N[(z * N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y + -1.0), $MachinePrecision] * N[(x * z), $MachinePrecision] + x), $MachinePrecision]]

Target

Original	96.2%
Target	99.8%
Herbie	97.7%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 1e5

   1. Initial program 97.0%

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

   2. Simplified 97.0%

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

      [Start] 97.0%	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      sub-neg [=>] 97.0%	\[ x \cdot \color{blue}{\left(1 + \left(-\left(1 - y\right) \cdot z\right)\right)} \]
      +-commutative [=>] 97.0%	\[ x \cdot \color{blue}{\left(\left(-\left(1 - y\right) \cdot z\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 97.0%	\[ x \cdot \left(\color{blue}{\left(1 - y\right) \cdot \left(-z\right)} + 1\right) \]
      sub-neg [=>] 97.0%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      +-commutative [=>] 97.0%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt1-in [<=] 97.0%	\[ x \cdot \left(\color{blue}{\left(\left(-z\right) + \left(-y\right) \cdot \left(-z\right)\right)} + 1\right) \]
      distribute-rgt-neg-in [<=] 97.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-\left(-y\right) \cdot z\right)}\right) + 1\right) \]
      associate-+l+ [=>] 97.0%	\[ x \cdot \color{blue}{\left(\left(-z\right) + \left(\left(-\left(-y\right) \cdot z\right) + 1\right)\right)} \]
      associate-+l+ [<=] 97.0%	\[ x \cdot \color{blue}{\left(\left(\left(-z\right) + \left(-\left(-y\right) \cdot z\right)\right) + 1\right)} \]
      distribute-rgt-neg-in [=>] 97.0%	\[ x \cdot \left(\left(\left(-z\right) + \color{blue}{\left(-y\right) \cdot \left(-z\right)}\right) + 1\right) \]
      distribute-rgt1-in [=>] 97.0%	\[ x \cdot \left(\color{blue}{\left(\left(-y\right) + 1\right) \cdot \left(-z\right)} + 1\right) \]
      +-commutative [<=] 97.0%	\[ x \cdot \left(\color{blue}{\left(1 + \left(-y\right)\right)} \cdot \left(-z\right) + 1\right) \]
      sub-neg [<=] 97.0%	\[ x \cdot \left(\color{blue}{\left(1 - y\right)} \cdot \left(-z\right) + 1\right) \]
      distribute-rgt-neg-in [<=] 97.0%	\[ x \cdot \left(\color{blue}{\left(-\left(1 - y\right) \cdot z\right)} + 1\right) \]
      *-commutative [=>] 97.0%	\[ x \cdot \left(\left(-\color{blue}{z \cdot \left(1 - y\right)}\right) + 1\right) \]
      distribute-rgt-neg-in [=>] 97.0%	\[ x \cdot \left(\color{blue}{z \cdot \left(-\left(1 - y\right)\right)} + 1\right) \]
      fma-def [=>] 97.0%	\[ x \cdot \color{blue}{\mathsf{fma}\left(z, -\left(1 - y\right), 1\right)} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(\left(y + -1\right) \cdot x\right) \cdot z + x} \]
      Step-by-step derivation

      [Start] 97.0%	\[ x \cdot \mathsf{fma}\left(z, y + -1, 1\right) \]
      fma-udef [=>] 97.0%	\[ x \cdot \color{blue}{\left(z \cdot \left(y + -1\right) + 1\right)} \]
      distribute-rgt-in [=>] 97.0%	\[ \color{blue}{\left(z \cdot \left(y + -1\right)\right) \cdot x + 1 \cdot x} \]
      *-commutative [=>] 97.0%	\[ \color{blue}{\left(\left(y + -1\right) \cdot z\right)} \cdot x + 1 \cdot x \]
      associate-*r* [<=] 96.5%	\[ \color{blue}{\left(y + -1\right) \cdot \left(z \cdot x\right)} + 1 \cdot x \]
      *-commutative [<=] 96.5%	\[ \left(y + -1\right) \cdot \color{blue}{\left(x \cdot z\right)} + 1 \cdot x \]
      associate-*r* [=>] 99.9%	\[ \color{blue}{\left(\left(y + -1\right) \cdot x\right) \cdot z} + 1 \cdot x \]
      *-un-lft-identity [<=] 99.9%	\[ \left(\left(y + -1\right) \cdot x\right) \cdot z + \color{blue}{x} \]

   if 1e5 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 94.7%

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

   2. Simplified 99.8%

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

      [Start] 94.7%	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      distribute-rgt-out-- [<=] 94.6%	\[ \color{blue}{1 \cdot x - \left(\left(1 - y\right) \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 94.6%	\[ \color{blue}{x} - \left(\left(1 - y\right) \cdot z\right) \cdot x \]
      cancel-sign-sub-inv [=>] 94.6%	\[ \color{blue}{x + \left(-\left(1 - y\right) \cdot z\right) \cdot x} \]
      +-commutative [=>] 94.6%	\[ \color{blue}{\left(-\left(1 - y\right) \cdot z\right) \cdot x + x} \]
      distribute-lft-neg-in [=>] 94.6%	\[ \color{blue}{\left(\left(-\left(1 - y\right)\right) \cdot z\right)} \cdot x + x \]
      associate-*l* [=>] 99.8%	\[ \color{blue}{\left(-\left(1 - y\right)\right) \cdot \left(z \cdot x\right)} + x \]
      fma-def [=>] 99.8%	\[ \color{blue}{\mathsf{fma}\left(-\left(1 - y\right), z \cdot x, x\right)} \]
      neg-sub0 [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{0 - \left(1 - y\right)}, z \cdot x, x\right) \]
      associate--r- [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{\left(0 - 1\right) + y}, z \cdot x, x\right) \]
      metadata-eval [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{-1} + y, z \cdot x, x\right) \]
      +-commutative [=>] 99.8%	\[ \mathsf{fma}\left(\color{blue}{y + -1}, z \cdot x, x\right) \]
      *-commutative [<=] 99.8%	\[ \mathsf{fma}\left(y + -1, \color{blue}{x \cdot z}, x\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 100000:\\ \;\;\;\;x - z \cdot \left(x \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 100000:\\ \;\;\;\;x - z \cdot \left(x \cdot \left(1 - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \end{array} \]

Alternative 2
Accuracy	96.0%
Cost	836

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

Alternative 3
Accuracy	64.1%
Cost	785

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{+239}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-9} \lor \neg \left(z \leq 1.38 \cdot 10^{-15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	64.5%
Cost	785

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+273}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{+219}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-9} \lor \neg \left(z \leq 1.38 \cdot 10^{-15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	64.4%
Cost	785

\[\begin{array}{l} t_0 := x \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+271}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{+238}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-9} \lor \neg \left(z \leq 1.38 \cdot 10^{-15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	98.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+23} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;z \cdot \left(x \cdot \left(y + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + y \cdot z\right)\\ \end{array} \]

Alternative 7
Accuracy	84.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+28}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(y + -1\right) \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 8
Accuracy	84.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.62 \cdot 10^{+106}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{+29}:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \]

Alternative 9
Accuracy	95.9%
Cost	576

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

Alternative 10
Accuracy	64.5%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-9} \lor \neg \left(z \leq 1.38 \cdot 10^{-15}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	38.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023263 
(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))))