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

Average Accuracy: 96.0% → 99.9%

Time: 10.5s

Precision: binary64

Cost: 8137

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-29} \lor \neg \left(t_0 \leq 10^{+226}\right):\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- 1.0 (* (- 1.0 y) z)))))
   (if (or (<= t_0 -1e-29) (not (<= t_0 1e+226)))
     (fma (+ y -1.0) (* x z) x)
     t_0)))

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 = x * (1.0 - ((1.0 - y) * z));
	double tmp;
	if ((t_0 <= -1e-29) || !(t_0 <= 1e+226)) {
		tmp = fma((y + -1.0), (x * z), x);
	} else {
		tmp = t_0;
	}
	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(x * Float64(1.0 - Float64(Float64(1.0 - y) * z)))
	tmp = 0.0
	if ((t_0 <= -1e-29) || !(t_0 <= 1e+226))
		tmp = fma(Float64(y + -1.0), Float64(x * z), x);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(x * N[(1.0 - N[(N[(1.0 - y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-29], N[Not[LessEqual[t$95$0, 1e+226]], $MachinePrecision]], N[(N[(y + -1.0), $MachinePrecision] * N[(x * z), $MachinePrecision] + x), $MachinePrecision], t$95$0]]

Target

Original	96.0%
Target	99.7%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < -9.99999999999999943e-30 or 9.99999999999999961e225 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z)))

   1. Initial program 92.4%

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

   2. Simplified 99.9%

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

      [Start] 92.4	\[ x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
      distribute-rgt-out-- [<=] 92.4	\[ \color{blue}{1 \cdot x - \left(\left(1 - y\right) \cdot z\right) \cdot x} \]
      *-lft-identity [=>] 92.4	\[ \color{blue}{x} - \left(\left(1 - y\right) \cdot z\right) \cdot x \]
      cancel-sign-sub-inv [=>] 92.4	\[ \color{blue}{x + \left(-\left(1 - y\right) \cdot z\right) \cdot x} \]
      +-commutative [=>] 92.4	\[ \color{blue}{\left(-\left(1 - y\right) \cdot z\right) \cdot x + x} \]
      distribute-lft-neg-in [=>] 92.4	\[ \color{blue}{\left(\left(-\left(1 - y\right)\right) \cdot z\right)} \cdot x + x \]
      associate-*l* [=>] 99.9	\[ \color{blue}{\left(-\left(1 - y\right)\right) \cdot \left(z \cdot x\right)} + x \]
      fma-def [=>] 99.9	\[ \color{blue}{\mathsf{fma}\left(-\left(1 - y\right), z \cdot x, x\right)} \]
      neg-sub0 [=>] 99.9	\[ \mathsf{fma}\left(\color{blue}{0 - \left(1 - y\right)}, z \cdot x, x\right) \]
      associate--r- [=>] 99.9	\[ \mathsf{fma}\left(\color{blue}{\left(0 - 1\right) + y}, z \cdot x, x\right) \]
      metadata-eval [=>] 99.9	\[ \mathsf{fma}\left(\color{blue}{-1} + y, z \cdot x, x\right) \]
      +-commutative [=>] 99.9	\[ \mathsf{fma}\left(\color{blue}{y + -1}, z \cdot x, x\right) \]
      *-commutative [<=] 99.9	\[ \mathsf{fma}\left(y + -1, \color{blue}{x \cdot z}, x\right) \]

   if -9.99999999999999943e-30 < (*.f64 x (-.f64 1 (*.f64 (-.f64 1 y) z))) < 9.99999999999999961e225

   1. Initial program 99.9%

      \[x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq -1 \cdot 10^{-29} \lor \neg \left(x \cdot \left(1 - \left(1 - y\right) \cdot z\right) \leq 10^{+226}\right):\\ \;\;\;\;\mathsf{fma}\left(y + -1, x \cdot z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \left(1 - y\right) \cdot z\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	1353

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 4 \cdot 10^{+303}\right):\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	1352

\[\begin{array}{l} t_0 := \left(1 - y\right) \cdot z\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \left(x \cdot \left(1 - y\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	84.5%
Cost	848

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot z\right)\\ t_1 := x - x \cdot z\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+40}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	66.6%
Cost	716

\[\begin{array}{l} t_0 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -5.15 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 5
Accuracy	66.4%
Cost	716

\[\begin{array}{l} \mathbf{if}\;z \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+104}:\\ \;\;\;\;z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-z\right)\\ \end{array} \]

Alternative 6
Accuracy	95.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -440000000 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x + z \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array} \]

Alternative 7
Accuracy	95.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -440000000:\\ \;\;\;\;x + z \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x - x \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 8
Accuracy	65.2%
Cost	521

\[\begin{array}{l} \mathbf{if}\;z \leq -100000000 \lor \neg \left(z \leq 1.12 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	39.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023160 
(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))))